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The Laws of Exponents

Jan 03, 2020

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The Laws of Exponents. Exponents. exponent. Power. base. 5 3 means 3 factors of 5 or 5 x 5 x 5. The Laws of Exponents:. Exponential form: The exponent of a power indicates how many times the base multiplies itself. n factors of x.

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Exponents exponent Power base 53 means 3 factors of 5 or 5 x 5 x 5

The Laws of Exponents: Exponential form:The exponent of a power indicates how many times the base multiplies itself. n factors of x

#1: Multiplying Powers (Product of Powers): If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS! So, I get it! When you multiply Powers, you add the exponents!

#2: Dividing Powers (Quotient of Powers):When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS! So, I get it! When you divide Powers, you subtract the exponents!

#3: Power of a Power:If you are raising a Power to an exponent, you multiply the exponents! So, when I take a Power to a power, I multiply the exponents

#5: Product Law of Exponents:If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent. So, when I take a Power of a Product, I apply the exponent to all factors of the product.

#6: Quotient Law of Exponents:If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent. So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.

#4: Negative Law of Exponents:If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent. So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent. Ha Ha! If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!

#5: Zero Law of Exponents:Any base powered by zero exponent equals one. So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.

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Presentation on theme: "Laws of Exponents: Multiplication."— Presentation transcript:

Simplifying a Variable Expression

Laws of Exponents: Dividing Monomials Division Rules for Exponents.

Multiplication Rules for Exponents

Properties of Exponents

Lesson #3- Exponent Laws

WARM UP 3 SIMPLIFY THE EXPRESSION ● (2 3 ) 2.

Multiplying and Dividing Monomials 4.3 Monomial: An expression that is either a: (1) numeral or constant, ex : 5 (2)a v ariable, ex: x (3)or a product.

Chapter 6 Polynomial Functions and Inequalities. 6.1 Properties of Exponents Negative Exponents a -n = –Move the base with the negative exponent to the.

Section 8.1.  Exponents are a short hand way to write multiplication  Examples: 4 ·4 = 4 2 4·4·4 = 4 3 4·4·x·x·x= 4 2 x 3 = 16 x 3.

P ROPERTIES OF E XPONENTS. P RE -T EST Q UESTION 1 y 3 y 4 A.) y 12 B.) y 7 C.) y -1 D.) y.

Laws of Exponents. Review An exponent tells you how many times the base is multiplied by itself. h 5 means h∙ h∙ h∙ h∙ h Any number to the zero power.

Secret Signal … on your chest “1” if true “2” if false 2 3 = (2)(2)(2)

Review of Properties of Exponents. a 0 = 1, a  0 Properties of Exponents Assume throughout your work that no denominator is equal to zero and that m.

Ch 8: Exponents B) Zero & Negative Exponents

California Standards AF2.2 Multiply and divide monomials; extend the process of taking powers and extracting roots to monomials when the latter results.

Evaluating Algebraic Expressions 4-4 Multiplying and Dividing Monomials Math humor: Question: what has variables with whole-number exponents and a bunch.

Chapter 8.1.  Lesson Objective: NCSCOS 1.01 – Write the equivalent forms of algebraic expressions to solve problems  Students will know how to apply.

6.1 - Polynomials. Monomial Monomial – 1 term Monomial – 1 term; a variable.

Example Divide 2y 2 – 6y + 4g – 8 by 2. 2y 2 – 6y + 4g y 2 – 6y + 4g Simply divide each term by 2 y 2 – 3y + 2g - 4.

Aim: How do we work on the expression with negative or zero exponent?

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5.5: Laws of Exponents

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• Page ID 19881

• David Arnold
• College of the Redwoods

In Chapter 1, section 1, we first introduced the definition of an exponent. For convenience, we repeat that definition.

In the exponential expression \(a^n\), the number \(a\) is called the base, while the number \(n\) is called the exponent.

Let \(a\) be any real number and let \(n\) be any whole number. If \(n \neq 0\), then:

\[a^{n}=\underbrace{a \cdot a \cdot a \cdots \cdot a}_{n \text { times }} \nonumber \]

That is, to calculate \(a^n\), write \(a\) as a factor \(n\) times. In the case where \(a \neq 0\), but \(n = 0\), then we define:

\[a^0 =1 \nonumber \]

For example, raising a number to the fifth power requires that we repeat the number as a factor five times (see Figure \(\PageIndex{1}\)).

\[\begin{aligned}(-2)^{5} &=(-2)(-2)(-2)(-2)(-2) \\ &=-32 \end{aligned} \nonumber \]

Raising a number to the fourth power requires that we repeat that number as a factor four times (see Figure \(\PageIndex{1}\)).

\[\begin{aligned}\left(-\dfrac{1}{2}\right)^{4} &=\left(-\dfrac{1}{2}\right)\left(-\dfrac{1}{2}\right)\left(-\dfrac{1}{2}\right)\left(-\dfrac{1}{2}\right) \\ &=\dfrac{1}{16} \end{aligned} \nonumber \]

As a final example, note that \(10^0 = 1\), but \(0^0\) is undefined (see Figure \(\PageIndex{2}\)).

For those who may be wondering why \(a^0 = 1\), provided \(a \neq 0\) , here is a nice argument. First, note that \(a^1 = a\), so:

\[a \cdot a^{0}=a^{1} \cdot a^{0} \nonumber \]

On the right, repeat the base and add the exponents.

\[a \cdot a^{0}=a^{1} \nonumber \]

Or equivalently:

\[a \cdot a^{0}=a \nonumber \]

Now, divide both sides by \(a\), which is permissible if \(a \neq 0\) .

\[\dfrac{a \cdot a^{0}}{a}=\dfrac{a}{a} \nonumber \]

Simplify both sides:

Multiplying With Like Bases

In the expression \(a^n\), the number \(a\) is called the base and the number \(n\) is called the exponent . Frequently, we’ll be required to multiply two exponential expressions with like bases, such as \(x^{3} \cdot x^{4}\). Recall that the exponent tells us how many times to write each base as a factor, so we can write:

\[\begin{aligned} x^{3} \cdot x^{4} &=(x \cdot x \cdot x) \cdot(x \cdot x \cdot x \cdot x) \\ &=x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \\ &=x^{7} \end{aligned} \nonumber \]

Note that we are simply counting the number of times that \(x\) occurs as a factor. First, we have three \(x’s\), then four \(x’s\), for a total of seven \(x’s\). However a little thought tells us that it is much quicker to simply add the exponents to reveal the total number of times \(x\) occurs as a factor.

\[\begin{aligned} x^{3} \cdot x^{4} &=x^{3+4} \\ &=x^{7} \end{aligned} \nonumber \]

The preceding discussion is an example of the following general law of exponents.

To multiply two exponential expressions with like bases, repeat the base and add the exponents.

\[a^{m} \cdot a^{n}=a^{m+n} \nonumber \]

Example \(\PageIndex{1}\)

Simplify each of the following expressions:

• \(y^{4} \cdot y^{8}\)
• \(2^{3} \cdot 2^{5}\)
• \((x+y)^{2}(x+y)^{7}\)

In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and add the exponents.

• \(\begin{aligned} y^{4} \cdot y^{8} &=y^{4+8} \\ &=y^{12} \end{aligned}\)
• \(\begin{aligned} 2^{3} \cdot 2^{5} &=2^{3+5} \\ &=2^{8} \end{aligned}\)
• \(\begin{aligned}(x+y)^{2}(x+y)^{7} &=(x+y)^{2+7} \\ &=(x+y)^{9} \end{aligned}\)

With a little practice, each of the examples can be simplified mentally. Repeat the base and add the exponents in your head: \(y^{4} \cdot y^{8}=y^{12}, 2^{3} \cdot 2^{5}=2^{8}\) and \((x+y)^{2}(x+y)^{7}=(x+y)^{9}\).

Exercise \(\PageIndex{1}\)

\(3^{4} \cdot 3^{2}\)

Example \(\PageIndex{2}\)

Simplify: \(\left(a^{6} b^{4}\right)\left(a^{3} b^{2}\right)\)

We’ll use the commutative and associative properties to change the order of operation, then repeat the common bases and add the exponents.

\[\begin{aligned} \left(a^{6} b^{4}\right)\left(a^{3} b^{2}\right) &= a^{6} b^{4} a^{3} b^{2} \quad \color {Red} \text { The associative property allows us to regroup in the order we prefer. } \\ &= a^{6} a^{3} b^{4} b^{2} \quad \color {Red} \text { The commutative property allows us to change the order of multiplication. } \\ &= a^{9} b^{6} \quad \color {Red} \text { Repeat the common bases and add the exponents. } \end{aligned} \nonumber \]

With practice, we realize that if all of the operators are multiplication, then we can multiply in the order we prefer, repeating the common bases and adding the exponents mentally: \(\left(a^{6} b^{4}\right)\left(a^{3} b^{2}\right)=a^{9} b^{6}\).

Exercise \(\PageIndex{2}\)

\(\left(x^{2} y^{6}\right)\left(x^{4} y^{3}\right)\)

\(x^{6} y^{9}\)

Example \(\PageIndex{3}\)

Simplify: \(x^{n+3} \cdot x^{3-2 n}\)

Again, we repeat the base and add the exponents.

\[\begin{aligned} x^{n+3} \cdot x^{3-2 n} &= x^{(n+3)+(3-2 n)} \quad \color {Red}\text { Repeat the base, add the exponents. } \\ &= x^{6-n} \quad \color {Red} \text { Simplify. Combine like terms. } \end{aligned} \nonumber \]

Exercise \(\PageIndex{3}\)

\(x^{5-n} \cdot x^{4 n+2}\)

\(x^{3 n+7}\)

Dividing With Like Bases

Like multiplication, we will also be frequently asked to divide exponential expressions with like bases, such as \(x^7/x^4\). Again, the key is to remember that the exponent tells us how many times to write the base as a factor, so we can write:

\[\begin{aligned} \dfrac{x^{7}}{x^{4}} &= \dfrac{x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x}{x \cdot x \cdot x \cdot x} \\ &= \dfrac{\not {x} \cdot \not {x} \cdot \not{x} \cdot \not {x} \cdot x \cdot x \cdot x}{\not {x} \cdot \not {x} \cdot \not {x} \cdot \not {x}} \\ &= x^3 \end{aligned} \nonumber\]

Note how we cancel four \(x’s\) in the numerator for four \(x’s\) in the denominator. However, in a sense we are “subtracting four \(x’s\)” from the numerator, so a faster way to proceed is to repeat the base and subtract the exponents, as follows:

\[\begin{aligned} \dfrac{x^{7}}{x^{4}} &=x^{7-4} \\ &=x^{3} \end{aligned} \nonumber \]

The preceding discussion is an example of the second general law of exponents.

How to Divide with Like Bases

To divide two exponential expressions with like bases, repeat the base and subtract the exponents. Given \(a \neq 0\),

\[\dfrac{a^{m}}{a^{n}}=a^{m-n} \nonumber \]

Note that the subtraction of the exponents follows the rule “top minus bottom.”

Here is another nice argument why \(a^0 = 1\), provided \(a \neq 0\). Start with:

\[\dfrac{a^{1}}{a^{1}}=1 \nonumber \]

Repeat the base and subtract the exponents.

\[a^{1-1}=1 \nonumber \]

\[a^0 = 1 \nonumber \]

Example \(\PageIndex{4}\)

• \(\dfrac{x^{12}}{x^{3}}\)
• \(\dfrac{5^{7}}{5^{7}}\)
• \(\dfrac{(2 x+1)^{8}}{(2 x+1)^{3}}\)

In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and subtract the exponents.

• \(\begin{aligned} \dfrac{x^{12}}{x^{3}} &=x^{12-3} \\ &=x^{9} \end{aligned}\)
• \(\begin{aligned} \dfrac{5^{7}}{5^{7}} &=5^{7-7} \\ &=5^{0} \\ &=1 \end{aligned}\)
• \(\begin{aligned} \dfrac{(2 x+1)^{8}}{(2 x+1)^{3}} &=(2 x+1)^{8-3} \\ &=(2 x+1)^{5} \end{aligned}\)

With a little practice, each of the examples can be simplified mentally. Repeat the base and subtract the exponents in your head: \(x^{12} / x^{3}=x^{9}, 5^{7} / 5^{4}=5^{3}\) and \((2 x+1)^{8} /(2 x+1)^{3}=(2 x+1)^{5}\).

Exercise \(\PageIndex{4}\)

\(\dfrac{4^{5}}{4^{3}}\)

Example \(\PageIndex{5}\)

Simplify: \(\dfrac{12 x^{5} y^{7}}{4 x^{3} y^{2}}\)

We first express the fraction as a product of three fractions, the latter two with a common base. In the first line of the following solution, note that if you multiply numerators and denominators of the three separate fractions, the product equals the original fraction on the left.

\[\begin{aligned} \dfrac{12 x^{5} y^{7}}{4 x^{3} y^{2}} &= \dfrac{12}{4} \cdot \dfrac{x^{5}}{x^{3}} \cdot \dfrac{y^{7}}{y^{2}} \quad \color {Red} \text { Break into a product of three fractions. } \\ &= 3 x^{5-3} y^{7-2} \quad \color {Red} \text { Simplify: } 12 / 4=3 . \text { Then repeat the common } \\ &= 3 x^{2} y^{5} \quad \color {Red} \text { Simplify. } \end{aligned} \nonumber \]

Exercise \(\PageIndex{5}\)

Simplify: \(\dfrac{15 a^{6} b^{9}}{3 a b^{5}}\)

\(5 a^{5} b^{4}\)

Example \(\PageIndex{6}\)

Simplify: \(\dfrac{x^{5 n-4}}{x^{3-2 n}}\)

Again, we repeat the base and subtract the exponents.

\[\begin{aligned} \dfrac{x^{5 n-4}}{x^{3-2 n}} &= x^{(5 n-4)-(3-2 n)} \quad \color {Red} \text { Repeat the base, subtract exponents. } \\ &= x^{5 n-4-3+2 n} \quad \color {Red} \text { Distribute the minus sign. } \\ &= x^{7 n-7} \quad \color {Red} \text { Simplify. Combine like terms. } \end{aligned} \nonumber \]

Exercise \(\PageIndex{6}\)

Simplify: \(\dfrac{x^{3 n-6}}{x^{n+2}}\)

\(x^{2 n-8}\)

Raising a Power to a Power

Suppose we have an exponential expression raised to a second power, such as \((x^2)3\). The second exponent tells us to write \(x^2\) as a factor three times:

\[\begin{aligned} \left(x^{2}\right)^{3} &= x^{2} \cdot x^{2} \cdot x^{2} \quad \color {Red} \text { Write } x^{2} \text { as a factor three times. } \\ &= x^{6} \quad \color {Red} \text { Repeat the base, add the exponents. } \end{aligned} \nonumber \]

Note how we added \(2 + 2 + 2\) to get \(6\). However, a much faster way to add “three twos” is to multiply: \(3 \cdot 2=6\). Thus, when raising a “power to a second power,” repeat the base and multiply the exponents, as follows:

\[\begin{aligned}\left(x^{2}\right)^{3} &=x^{2 \cdot 3} \\ &=x^{6} \end{aligned} \nonumber \]

The preceding discussion gives rise to the following third law of exponents.

When raising a power to a power, repeat the base and multiply the exponents. In symbols: \[(a^m)^n = a^{mn} \nonumber \] Note that juxtaposing two variables, as in \(mn\), means “\(m\) times \(n\).”

Example \(\PageIndex{7}\)

• \(\left(z^{3}\right)^{5}\)
• \(\left(7^{3}\right)^{0}\)
• \(\left[(x-y)^{3}\right]^{6}\)

In each example we are raising a power to a power. Hence, in each case, we repeat the base and multiply the exponents.

• \(\begin{aligned}\left(z^{3}\right)^{5} &=z^{3.5} \\ &=z^{15} \end{aligned}\)
• \(\begin{aligned}\left(7^{3}\right)^{0} &=7^{3.0} \\ &=7^{0} \\ &=1 \end{aligned}\)
• \(\begin{aligned}\left[(x-y)^{3}\right]^{6} &=(x-y)^{3 \cdot 6} \\ &=(x-y)^{18} \end{aligned}\)

With a little practice, each of the examples can be simplified mentally. Repeat the base and multiply the exponents in your head: \(\left(z^{3}\right)^{5}=z^{15},\left(7^{3}\right)^{4}=7^{12}\) and \(\left[(x-y)^{3}\right]^{6}=(x-y)^{18}\).

Exercise \(\PageIndex{7}\)

Simplify: \(\left(2^{3}\right)^{4}\)

Example \(\PageIndex{8}\)

Simplify: \(\left(x^{2 n-3}\right)^{4}\)

Again, we repeat the base and multiply the exponents.

\[\begin{aligned} \left(x^{2 n-3}\right)^{4} &= x^{4(2 n-3)} \quad \color {Red} \text { Repeat the base, multiply exponents. } \\ &= x^{8 n-12} \quad \color {Red} \text { Distribute the } 4 . \end{aligned} \nonumber \]

Exercise \(\PageIndex{8}\)

Simplify: \(\left(a^{2-n}\right)^{3}\)

\(a^{6-3 n}\)

Raising a Product to a Power

We frequently have need to raise a product to a power, such as \((xy)^3\). Again, remember the exponent is telling us to write \(xy\) as a factor three times, so:

\[\begin{aligned} (x y)^{3} &=(x y)(x y)(x y) \quad \color {Red} \text {Write } xy \text { as a factor three times.}\\ &=x y x y x y \quad \color {Red} \text {The associative property allows us to group as we please.}\\ &=x x x y y y \quad \color {Red} \text {The commutative property allows us to change the order as we please.}\\ &=x^{3} y^{3} \quad \color {Red} \text {Invoke the exponent definition: } x x x=x^{3} \text { and } y y y=y^{3} \end{aligned} \nonumber \]

However, it is much simpler to note that when you raise a product to a power, you raise each factor to that power. In symbols: \[(xy)^3 = x^3y^3 \nonumber \] The preceding discussion leads us to a fourth law of exponents.

To raise a product to a power, raise each factor to that power. In symbols:

\[(ab)^n = a^nb^n \nonumber \]

Example \(\PageIndex{9}\)

• \((y z)^{5}\)
• \((-2 x)^{3}\)
• \((-3 y)^{2}\)

In each example we are raising a product to a power. Hence, in each case, we raise each factor to that power.

• \((y z)^{5}=y^{5} z^{5}\)
• \(\begin{aligned}(-2 x)^{3} &=(-2)^{3} x^{3} \\ &=-8 x^{3} \end{aligned}\)
• \(\begin{aligned}(-3 y)^{2} &=(-3)^{2} y^{2} \\ &=9 y^{2} \end{aligned}\)

With a little practice, each of the examples can be simplified mentally. Raise each factor to the indicated power in your head: \((y z)^{5}=y^{5} z^{5},(-2 x)^{3}=-8 x^{3}\) and \((-3 y)^{2}=9 y^{2}\)

Exercise \(\PageIndex{9}\)

Simplify: \((-2 b)^{4}\)

\(16 b^{4}\)

When raising a product of three factors to a power, it is easy to show that we should raise each factor to the indicated power. For example, \((a b c)^{3}=a^{3} b^{3} c^{3}\). In general, this is true regardless of the number of factors. When raising a product to a power, raise each of the factors to the indicated power.

Example \(\PageIndex{10}\)

Simplify: \(\left(-2 a^{3} b^{2}\right)^{3}\)

Raise each factor to the third power, then simplify.

\[\begin{aligned} \left(-2 a^{3} b^{2}\right)^{3} &= (-2)^{3}\left(a^{3}\right)^{3}\left(b^{2}\right)^{3} \quad \color {Red} \text { Raise each factor to the third power. } \\ &= -8 a^{9} b^{6} \quad \color {Red} \text {Simplify: } (-2)^{3}=8 \text {. In the remaining factors, raising a power to a power requires that we multiply the exponents. } \end{aligned} \nonumber \]

Exercise \(\PageIndex{10}\)

Simplify: \(\left(-3 x y^{4}\right)^{5}\)

\(-243 x^{5} y^{20}\)

Example \(\PageIndex{11}\)

Simplify: \(\left(-2 x^{2} y\right)^{2}\left(-3 x^{3} y\right)\)

In the first grouped product, raise each factor to the second power.

\[\begin{aligned} \left(-2 x^{2} y\right)^{2}\left(-3 x^{3} y\right) &=\left((-2)^{2}\left(x^{2}\right)^{2} y^{2}\right)\left(-3 x^{3} y\right) \quad \color {Red} \text { Raise each factor in the first grouped product to the second power.} \\ &=\left(4 x^{4} y^{2}\right)\left(-3 x^{3} y\right) \quad \color {Red} \text { Simplify: } (-2)^{2}=4 \text { and } \left(x^{2}\right)^{2}=x^{4} \end{aligned} \nonumber \]

The associative and commutative property allows us to multiply all six factors in the order that we please. Hence, we’ll multiply \(4\) and \(−3\), then \(x^4\) and \(x^3\), and \(y^2 and y, in that order. In this case, we repeat the base and add the exponents.

\[\begin{aligned} &=-12x^7y^3 \quad \color {Red} \text { Simplify: } (4)(-3)=-12. \text { Also, } x^{4}x^3=x^{7} \text { and } y^2y = y^3\end{aligned} \nonumber\]

Exercise \(\PageIndex{11}\)

Simplify: \(\left(-a^{3} b^{2}\right)^{3}\left(-2 a^{2} b^{4}\right)^{2}\)

\(-4 a^{13} b^{14}\)

Raising a Quotient to a Power

Raising a quotient to a power is similar to raising a product to a power. For example, raising \((x/y)^3\) requires that we write \(x/y\) as a factor three times.

\[\begin{aligned}\left(\frac{x}{y}\right)^{3} &=\frac{x}{y} \cdot \frac{x}{y} \cdot \frac{x}{y} \\ &=\frac{x \cdot x \cdot x}{y \cdot y \cdot y} \\ &=\frac{x^{3}}{y^{3}} \end{aligned} \nonumber \]

However, it is much simpler to realize that when you raise a quotient to a power, you raise both numerator and denominator to that power. In symbols:

\[\left(\dfrac{x}{y}\right)^{3}=\frac{x^{3}}{y^{3}} \nonumber \]

This leads to the fifth and final law of exponents.

To raise a quotient to a power, raise both numerator and denominator to that power. Given \(b \neq 0\),

\[\left(\dfrac{a}{b}\right)^{n}=\dfrac{a^{n}}{b^{n}} \nonumber \]

Example \(\PageIndex{12}\)

• \(\left(\dfrac{2}{3}\right)^{2}\)
• \(\left(\dfrac{x}{3}\right)^{3}\)
• \(\left(-\dfrac{2}{y}\right)^{4}\)

In each example we are raising a quotient to a power. Hence, in each case, we raise both numerator and denominator to that power.

• \(\begin{aligned}\left(\dfrac{2}{3}\right)^{2} &=\dfrac{2^{2}}{3^{2}} \\ &=\dfrac{4}{9} \end{aligned}\)
• \(\begin{aligned}\left(\dfrac{x}{3}\right)^{3} &=\dfrac{x^{3}}{3^{3}} \\ &=\dfrac{x^{3}}{27} \end{aligned}\)
• \(\begin{aligned}\left(-\dfrac{2}{y}\right)^{4} &=\dfrac{2^{4}}{y^{4}} \\ &=\dfrac{16}{y^{4}} \end{aligned}\)

Note that in example (c), raising a negative base to an even power produces a positive result. With a little practice, each of the examples can be simplified mentally. Raise numerator and denominator to the indicated power in your head: \((2 / 3)^{2}=4 / 9,(x / 3)^{3}=x^{3} / 27,\) and \((-2 / y)^{4}=16 / y^{4}\)

Exercise \(\PageIndex{12}\)

Simplify: \(\left(\dfrac{5}{4}\right)^{3}\)

\(\dfrac{125}{64}\)

Example \(\PageIndex{13}\)

Simplify: \(\left(\dfrac{2 x^{5}}{y^{3}}\right)^{2}\)

Raise both numerator and denominator to the second power, then simplify:

\[\begin{aligned} \left(\dfrac{2 x^{5}}{y^{3}}\right)^{2}=\dfrac{\left(2 x^{5}\right)^{2}}{\left(y^{3}\right)^{2}} \quad \color {Red} \text {Raise numerator and denominator to the second power.} \end{aligned} \nonumber\]

In the numerator, we need to raise each factor of the product to the second power. Then we need to remind ourselves that when we raise a power to a power, we multiply the exponents.

\[\begin{aligned} &= \dfrac{2^{2}\left(x^{5}\right)^{2}}{\left(y^{3}\right)^{2}} \quad \color {Red} \text {Raise each factor in the numerator and denominator to the second power.} \\ &= \dfrac{4 x^{10}}{y^{6}} \quad \color {Red} \text { Simplify: } 2^{2}=4,\left(x^{5}\right)^{2}=x^{10} \text {, and } (y^3)^2=y^6 \end{aligned} \nonumber\]

Exercise \(\PageIndex{13}\)

Simplify: \(\left(\dfrac{a^{4}}{3 b^{2}}\right)^{3}\)

\(\dfrac{a^{12}}{27 b^{6}}\)

Laws of exponents

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• 1. NOTES Algebra 1 Mr. Saucedo
• 2. REVIEW  Recall: A number in exponential form has a base and an exponent. The exponent indicates how many times the base is used as a factor.  In its exponential form: ab , a is the BASE and b is the EXPONENT.
• 3. NOTES The laws of exponents show how to __SIMPLIFY___ expressions in exponential form.
• 4. NOTES In the next few slides, a and b are real numbers and m and n are integers.
• 5. PRODUCT OF POWERS  To multiply powers with the same base, add the exponents. am ∙an = am+n
• 6. POWER OF A POWER  To raise a number in exponential form to a power, multiply the exponents. (am )n = am∙n
• 7. POWER OF A PRODUCT  To find a power of a product, find the power of each factor and multiply. (ab)m = am bm
• 8. POWER OF ZERO  Any nonzero number raised to the power of zero is one. a0 = 1
• 9. QUOTIENT OF POWERS  To find the quotient of powers with the same base, subtract the exponents. am = am-n an
• 10. POWER OF A QUOTIENT To raise a quotient to a power, raise both the numerator and denominator to that power. (a/b)m = am /bm
• 11. RECIPROCALS  To change a sign of an exponent, move the expression to the denominator of a fraction, or to the numerator. a-n = 1/an 1/a-n = an
• 12. GUIDED PRACTICE  Simplify: 1. (x3 y2 z0 )3 2. (x4 y5 /xy2 )2
• 13. GUIDED PRACTICE 3. What is the value of the expression 6-3 ? 4. What is the result when a4 b-3 is divided by a-2 b2
• 14. CHECKPOINT Simplify: 1. (4m3 n3 )2 2. x3 y-2 /y4 3. (5a)2 b5 /(3a4 b2 ) 4. (3b2 x-2 y6 )0
• 15. INDEPENDENT PRACTICE  Workbook, page 16. Problems 1 - 9.

Laws of Exponents

Exponents are also called Powers or Indices

The exponent of a number says how many times to use the number in a multiplication.

In this example: 8 2 = 8 × 8 = 64

Try it yourself:

So an Exponent saves us writing out lots of multiplies!

Example: a 7

a 7 = a × a × a × a × a × a × a = aaaaaaa

Notice how we wrote the letters together to mean multiply? We will do that a lot here.

Example: x 6 = xxxxxx

The key to the laws.

Writing all the letters down is the key to understanding the Laws

Example: x 2 x 3 = (xx)(xxx) = xxxxx = x 5

Which shows that x 2 x 3 = x 5 , but more on that later!

So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.

All you need to know ...

The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas :

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Here are the Laws (explanations follow):

Laws Explained

The first three laws above ( x 1 = x , x 0 = 1 and x -1 = 1/x ) are just part of the natural sequence of exponents. Have a look at this:

Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

The law that x m x n = x m+n

With x m x n , how many times do we end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.

So, x 2 x 3 = x (2+3) = x 5

The law that x m /x n = x m-n

Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we are dividing), for a total of "m-n" times.

Example: x 4 /x 2 = (xxxx) / (xx) = xx = x 2

So, x 4 /x 2 = x (4-2) = x 2

(Remember that x / x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)

This law can also show you why x 0 =1 :

Example: x 2 /x 2 = x 2-2 = x 0 =1

The law that (x m ) n = x mn.

First you multiply "m" times. Then you have to do that "n" times , for a total of m×n times.

Example: (x 3 ) 4 = (xxx) 4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x 12

So (x 3 ) 4 = x 3×4 = x 12

The law that (xy) n = x n y n

To show how this one works, just think of re-arranging all the "x"s and "y"s as in this example:

Example: (xy) 3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x 3 y 3

The law that (x/y) n = x n /y n.

Similar to the previous example, just re-arrange the "x"s and "y"s

Example: (x/y) 3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x 3 /y 3

The law that   x m/n  = n √ x m  = ( n √ x ) m.

OK, this one is a little more complicated!

I suggest you read Fractional Exponents first, so this makes more sense.

Anyway, the important idea is that:

x 1/ n = The n- th Root of x

And so a fractional exponent like 4 3/2 is really saying to do a cube (3) and a square root (1/2), in any order.

Just remember from fractions that m/n = m × (1/n) :

Example: x ( m n )  =  x (m × 1 n )  =  (x m ) 1/n  =  n √ x m

The order does not matter, so it also works for m/n = (1/n) × m :

Example: x ( m n )  =  x ( 1 n × m)  =  (x 1/n ) m  =  ( n √ x ) m

Exponents of exponents ....

What about this example?

We do the exponent at the top first , so we calculate it this way:

And That Is It!

If you find it hard to remember all these rules, then remember this:

you can work them out when you understand the three ideas near the top of this page:

• The exponent says how many times to use the number in a multiplication
• A negative exponent means divide
• A fractional exponent like 1/n means to take the nth root :   x ( 1 n ) = n √ x  

Oh, One More Thing ... What if x = 0?

The strange case of 0 0.

There are different arguments for the correct value of 0 0

0 0 could be 1, or possibly 0, so some people say it is really "indeterminate":

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Presentation Listings

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
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Laws of Exponents

• Rational Exponents
• Laws of Exponents for Real Numbers
• Fractional Exponents
• How to Multiply and Divide Exponents?
• Laws of Logarithms
• How do you express large numbers?
• Adding and Subtracting Exponents
• What is 125 in the exponential form?
• How to express a number as a power?
• Solve the expression: (7/3) (-3)
• What is 3 to the 2nd power mean?
• What is 6 to the 3rd power?
• What is 3 to the 2nd Power?
• Laws of Exponents & Use of Exponents to Express Small Numbers in Standard Form - Exponents and Powers | Class 8 Maths
• Find x in the following expression: (11/9) 3 × (9/11) 6 = (11/9) 2x-1
• Is 2 33 = 213 3 ?
• Evaluate (√4)-3
• Evaluate a 2 × a 3 × a -5
• What are Numbers?
• Propositional Logic
• Pigeonhole Principle
• L U Decomposition
• Set Theory - Definition, Types of Sets, Symbols & Examples
• Predicates and Quantifiers
• Binary to Decimal Converter
• Number System in Maths
• Puzzle 26 | (Know Average Salary without Disclosing Individual Salaries)

Laws of Exponents: Exponents are a way of representing very large or very small numbers. Exponent rules are the laws of the exponents that are used to solve various exponents’ problems. The multiplication, division, and other operations on exponents can be achieved using these laws of exponents. There are different rules of exponents also called laws of exponents in Mathematics and all these laws are added in the article below.

In this article, we will learn about Exponents Definition, Laws of Exponents, Laws of Exponents Examples, and others in detail.

Table of Content

Exponents Definition

What are exponent rules, what are laws of exponents, laws of exponents and logarithms, table: laws of exponents, exponent rules examples.

When a number is raised to some power then the power on the base number is called Exponent. Exponent simply means a base number is multiplied by itself equal to the power mentioned on it.

For Example, if we say P n this means P is multiplied by itself ‘n’ several times. It can be expanded as P×P×P×P×P×P . . . n times. Let’s say, 5 3 = 5 × 5 × 5 = 125; the equation is read as “five to the power of three.” 

If the exponent is 2 then it is also known as “squared,” whereas if the exponent is 3 it is known as “cubed.” When calculating the area, the term ‘squared’ is used because we multiply the length (m/cm) two times and in the case of volume the term ‘cubed’ is used as we multiply the length (unit = m/cm) three times.

Exponent helps us to write very large as well as very small quantities. For instance, we can write large quantities such as the Mass of the Earth which is 5.97219×10 24 kg as well as very small quantities such as the Mass of the Electron which is 9.1×10 -31 kg.

Read in Detail: Exponents: Definition, Formulas, Laws, and Examples

Exponent rules are the rules that are used to solve exponent’s problems. Suppose we are given two exponents a m and a n and we have to find the product of the two exponents then we use the concept of exponents rule or product of exponents rule, i.e.

a m × a n = a (m+n)

Various other rules are used to solve exponent problems. These rules are called the exponents rule.

Laws of Exponents are the set of rules that help us to solve arithmetic problems in an easy manner. Since at times, we can get large exponents that make multiplication lengthy then with the help of laws of exponents, we can solve the problems easily and in time bound manner.

Following are the seven Laws of Exponents that we must know to solve arithmetic problems involving exponents:

Product of Powers Rule

Quotient of powers rule.

• Power of a Powers Rule

Power of a Quotient Rule

Zero power rule, negative exponent rule.

In the Product of Powers Rule , if two numbers with the same bases and different exponents are multiplied then exponents of the base are added to find the product. It is represented as x m ×x n = x (m+n)

Example: 5 2 × 5 3 =?

Keep the base values the same because they’re both five, and then add the exponents together (2+3). 5 2 × 5 3 = 5 2+3 = 5 5 To get the answer, multiply five by itself five times. 5 5 = 5 × 5 × 5 × 5 × 5 = 3125

In Quotient of Powers Rule , if two numbers with the same bases and different exponents are divided then the exponents of the base are subtracted to find the quotient. It is represented as x a ÷x b = x (a-b)

Example: 4 5 ÷ 4 3 =?

4 5 ÷ 4 3 =? Because both bases in this equation are four, they remain the same. Then subtract the divisor from the dividend using the exponents. 4 5 ÷ 4 3 = 4 5-3 = 4 2 Finally, if necessary, simplify the equation. 4 2 = 4 × 4 = 16

Power of a Power Rule 

In Power of a Power Rule , if a number raised to some power is again raised to some power then the two powers will be multiplied. It is represented as (x m ) n = x m×n

Example: (2 3 ) 2 =?

(2 3 ) 2 =? Multiply the exponents together in equations like the one above while keeping the base constant. 2 3×2 = 2 6 However , we have to keep in mind that ((2^3)^2 ~\neq~2^{3^2} as (2 3 ) 2 = 2 6 but 2^{3^2} = 2^9 as only exponent 3 is again raised to exponent 2 and not the whole number including base. 

Power of a Product Rule

In Power of a Product Rule , two different bases are raised to the same power are multiplied, then, bases are multiplied and power is common to the product of the bases. It is represented as (x m × y m ) = (xy) m . If the given question is (xy) m then distribute the exponent to each portion of the base when multiplying any base by an exponent, hence (xy) m = (x m × y m )

Example: 2 3 ×3 3 =?

Since the bases are different and the power is same then multiply the bases and raise it to the common power. Therefore, 2 3 ×3 3 =(2×3) 3 = 6 3 = 216

Example: (2×3) 3 =? 

In this case separate the same power to individual bases. Hence, (2×3) 3 = 2 3 ×3 3 = 8×27 = 216

In Power of a Quotient Rule , if two different bases with the same power are divided then the result is the quotient of the bases raised to the same power. This is represented as x m /y m = (x/y) m . In this case, vice versa is also true i.e. if the both numerator and denominator are raised to the same power then power is distributed to both numerator and denominator individually. It can be represented as (x/y) m =  x m /y m

Example: Simplify 6 4 /3 4 .

In this case, find the quotient of the bases and raise common power to it. 6 4 /3 4 = (6/3) 4 = 2 4 = 16

Example: Simplify (6/3) 4 .

In this case, distribute the power 4 to both the numerator and denominator. (6/3) 4 = 6 4 /3 4 = (6×6×6×6)/(3×3×3×3) = 2×2×2×2 = 16

In Zero Power Rule , if any base is raised to power zero, then the result will be 1. This can be represented as x 0 = 1. Zero Power rule can be understood from the following description

Suppose we have to prove x 0 = 1. x 0 = x n-n , where (0 = n-n) From the Quotient of Power Rule, we know that if the base are same then we subtract the exponents while finding the quotient; the vice versa of Quotient of Power Rule also holds true.  ⇒ x n-n = x n /x n = 1 Hence, x 0 = 1. 

Let’s consider an example for better understanding of the law.

Example: (1001) 0 =?

As per Zero Power Rule, any number raised to power zero results the value 1. (1001) 0 = 1

In Negative Exponent Rule , if a number is raised to negative interest then we convert the base to its reciprocal, and the power is changed to positive. The vice versa is also true i.e. if the exponent is positive and if the base is converted to its reciprocal then the exponent is changed to the negative value. It can be represented as (x/y) -m = (y/x) m

Example: (2/3) -2 =?

Since, the exponent is negative the base is converted to its reciprocal. ⇒ (2/3) -2 = (3/2) 2 = 3 2 /2 2 = 9/4

Fractional Exponent Rule (Laws of Exponents with Fractions)

Fractional Exponent rule is a rule that is used to solve fractional exponents or the exponents that are in fractional form. An exponent in fractional form is written as a 1/n and is read as nth root of a. It is also represented as,

a 1/n = n √(a)

Here, a is the base of exponent and 1/n is the exponent in fractional form.

For example, simplify (8) 1/3

= (8) 1/3 = ∛(8) = ∛(2×2×2) = 2

Other Rules of Exponents

Apart from the above Seven Rules of Exponents, the following are some other Rules of Law of Exponents that we need to keep in mind while solving the questions of exponents.

If a negative number is raised to even number power then the result will be positive and if a negative number is raised to odd number power then the result is always negative. For example (-2) 4 = 16 and (-2) 5 = -32. If 1 is raised to any power then the result will be always 1. For example, 1 3 = 1, 1 1001 = 1. If any number except 1 is raised to power infinity then the result will be infinity. 2 ∞ = ∞

The Laws of Exponents and the Logarithim Rules are two rules that are used to solve various mathematical problems and these rules are added in the table below.

The above-mentioned 7 Laws of Exponents are summarized in the following table:

People Also Read:

Negative Exponents How to multiply and divide exponents Adding and Subtracting of Exponents Laws of Exponents for Real Numbers

Example 1: What is the simplification of 7 3 × 7 1 ?

7 3 × 7 1 = 7 3+1 = 7 4

Example 2: Simplify and find the value of 10 2 /5 2 .

We can write the given expression as; 10 2 /5 2 = (10/5) 2 = 2 2 = 4

Example 3: Find the value of (256) 3/4

(256) 3/4 = (4 4 ) 3/4 = 4 4×(3/4) = 4 3 = 64

Example 4: Find the value of 7 -3

7 -3 = (1/7) 3 = 1 3 /7 3 = 1/343

Example 5: Find the value of x if 125 = 25/5 x

We have 125 = 25/5 x  ⇒ 5 3 = 5 2 /5 x  ⇒ 5 3 = 5 2-x Now the quantity is the same on both sides and bases are also the same, hence, exponents will also be the same.  ⇒ 3 = 2-x ⇒ x = 2-3 = -1

FAQs on Exponent Rules

What are exponents in maths.

Exponent refers to the power raised on a number which basically means the number is multiplied by itself to the number of times equal to the power.

What is the Product of Powers Rule?

Product of Power rule states that when two numbers with the same base are raised to different then the product of the number will have the power equal to the sum of the powers of both numbers. It is given as  x m × x n = x (m+n)

What is Power of Power Rule?

Power of Power rule states that when a number is raised to some power and the whole number including the first power is again raised to some power, then the two powers are multiplied.

What is Zero Exponent Rule?

Zero Exponent Rule states that if any number is raised to power 0 then it will result 1. It is given as X 0 = 1.

What is the Value of 0 0 ?

The value of 0 0 is not defined in mathematics.

What are 8 Laws of Exponents?

The 8 Laws of Exponents are, Product Law: a m × a n = a m+n Quotient Law: a m /a n = a m-n Zero Exponent Law: a 0 = 1 Identity Exponent Law: a 1 = a Power of a Power: (a m ) n = a mn Power of a Product: (ab) m = a m b m Power of a Quotient: (a/b) m = a m /b m Negative Exponents Law: a -m = 1/a m

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The following outline provides a high-level overview of the FTC’s proposed final rule :

• Specifically, the final rule provides that it is an unfair method of competition—and therefore a violation of Section 5 of the FTC Act—for employers to enter into noncompetes with workers after the effective date.
• Fewer than 1% of workers are estimated to be senior executives under the final rule.
• Specifically, the final rule defines the term “senior executive” to refer to workers earning more than $151,164 annually who are in a “policy-making position.”
• Reduced health care costs: $74-$194 billion in reduced spending on physician services over the next decade.
• New business formation: 2.7% increase in the rate of new firm formation, resulting in over 8,500 additional new businesses created each year.
• This reflects an estimated increase of about 3,000 to 5,000 new patents in the first year noncompetes are banned, rising to about 30,000-53,000 in the tenth year.
• This represents an estimated increase of 11-19% annually over a ten-year period.
• The average worker’s earnings will rise an estimated extra $524 per year. 

The Federal Trade Commission develops policy initiatives on issues that affect competition, consumers, and the U.S. economy. The FTC will never demand money, make threats, tell you to transfer money, or promise you a prize. Follow the  FTC on social media , read  consumer alerts  and the  business blog , and  sign up to get the latest FTC news and alerts .

Contact Information

Media contact.

Victoria Graham Office of Public Affairs 415-848-5121

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Laws of Exponents - PowerPoint PPT Presentation

Laws of Exponents

Laws of exponents whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them. those operations are ... – powerpoint ppt presentation.

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