1.4.2: The Power Rules for Exponents

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Dec 14, 2021
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- 1.4.1.1: 1.4.1 and 1.4.2 Exercises
- 1.4.2.1: Exercises

Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
Rio Hondo College

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1.4.3 Learning Objectives

Use the Power Rule for Powers to simplify expressions involving exponents
Use the Power Rule for Products to simplify expressions involving exponents
Use the Power Rule for Quotients to simplify expressions involving exponents

The Power Rule for Powers

The following examples suggest a rule for raising a power to a power:

$(a^2)^3 = a^2 \cdot a^2 \cdot a^2$
Using the product rule we get:
$(a^2)^3 = a^{2+2+2}$
$(a^2)^3 = a^{3 \cdot 2}$
$(a^2)^3 = a^6$

$(x^9)^4 = x^9 \cdot x^9 \cdot x^9 \cdot x^9$
$(x^9)^4 = x^{9+9+9+9}$
$(x^9)^4 = x^{4 \cdot 9}$
$(x^9)^4 = x^{36}$

Power Rule for Powers

If $x$ is a real number and $n$ and $m$ are natural numbers,
$(x^n)^m = x^{n \cdot m}$

To raise a power to a power, multiply the exponents.

Example 1

Simplify each expression using the power rule for powers. All exponents are natural numbers.

$(7^3)^4 = 7^{3 \cdot 4} = 7^{12}$
$(y^5)^3 = y^{5 \cdot 3} = y^{15}$
$(d^{20})^6 = d^{20 \cdot 6} = d^{120}$

Try It Now 1

Simplify the expression using the power rule for powers.

$(8^5)^4$

Answer: $8^{20}$

Try It Now 2

Simplify the expression using the power rule for powers.

$(y^7)^7$

Answer: $y^{49}$

The Power Rule for Products

The following examples suggest a rule for raising a product to a power:

$\begin{aligned} &(a b)^{3}=a b \cdot a b \cdot a b \text { Use the commutative property of multiplication. }\\ &\begin{array}{l} =a a a b b b \\ =a^{3} b^{3} \end{array} \end{aligned}$

$\begin{aligned} (x y)^{5} &=x y \cdot x y \cdot x y \cdot x y \cdot x y \\ &=x x x x x \cdot \text { yyyyy } \\ &=x^{5} y^{5} \end{aligned}$

$\begin{aligned} (4 x y \mathrm{z})^{2} &=4 x y z \cdot 4 x y z \\ &=4 \cdot 4 \cdot x x \cdot y y \cdot z z \\ &=16 x^{2} y^{2} z^{2} \end{aligned}$

Power Rule for Products

If $x$ and $y$ are real numbers are $n$ is a natural number,
$(xy)^n = x^ny^n$

To raise a product to a power, apply the exponent rule to each and every factor

Example 2

Make use of either the power rule for products or power rule for powers to simplify each expression.

$(2b)^7 = 2^7b^7$
$(axy)^4 = a^4x^4y^4$
$(3ab)^2 = 3^2a^2b^2 = 9a^2b^2$

Don't forget to apply the exponent to the 3!

Example 3

Make use of both the power rule for products and power rule for powers to simplify each expression.

$(ab^3)^2 = a^2(b^3)^2 = a^2b^6$

We used two rules here. First, the power rule for products. Second, the power rule for powers.
$(7a^4b^2c^8)^2 = 7^2(a^4)^2(b^2)^2(c^8)^2 = 49a^8b^4c^{16}$
$[2(x+1)^4]^6 = 2^6(x+1)^{24} = 64(x+1)^{24}$

Example 4

If $6a^3c^7 \not = 0$ , then $(6a^3c^7)^0 = 1$ . Recall that $x^0 = 1$ for $x \not = 0$

The Power Rule for Quotients

The following example suggests a rule for raising a quotient to a power.

$(\dfrac{a}{b})^3 = \dfrac{a}{b} \cdot \dfrac{a}{b} \cdot \dfrac{a}{b} = \dfrac{a \cdot a \cdot a}{b \cdot b \cdot b} = \dfrac{a^3}{b^3}$

Power Rule for Quotients

If $x$ and $y$ are real numbers and $n$ is a natural number,

$(\dfrac{x}{y})^n = \dfrac{x^n}{y^n}, y \not = 0$

To raise a quotient to a power, distribute the exponent to both the numerator and denominator.

Example 5

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. All exponents are natural numbers.

$(\dfrac{x}{y})^6 = \dfrac{x^6}{y^6}$
$(\dfrac{2x}{b})^4 = \dfrac{(2x)^4}{b^4} = \dfrac{2^4x^4}{b^4} = \dfrac{16x^4}{b^4}$
$(\dfrac{a^3}{b^5})^7 = \dfrac{(a^3)^7}{(b^5)^7} = \dfrac{a^21}{b^35}$

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Power Rule for Powers

Example 1

Power Rule for Products

Example 2

Example 3

Example 4

Power Rule for Quotients

Example 5

The Power Rule for Powers

Power Rule for Powers

Example 1

The Power Rule for Products

Power Rule for Products

Example 2

Example 3

Example 4

The Power Rule for Quotients

Power Rule for Quotients

Example 5

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