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Algebra II : Distributing Exponents (Power Rule)

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Example Questions

Example Question #2 : Simplifying Exponents

What is the largest positive integer, , such that 2^z is a factor of 16^4 ?

Possible Answers:

Correct answer:

Explanation:

16^4 = (2^4)^4 = 2^1^6 . Thus, is equal to 16.

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Example Question #1 : Simplifying Exponents

Simplify the expression:

$(3x^4y^2)(4xy^2)^{-3}$

Possible Answers:

$\frac{3x}{64y^3}$

$\frac{3x}{64y^4}$

Cannot be simplified

$\frac{3x}{16y^3}$

Correct answer:

$\frac{3x}{64y^4}$

Explanation:

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

$(3x^{4}y^2)(4xy^2)^{-3}= (3xy^2)(4^{-3}x^{-3}y^{-6})$

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

$\frac{3x^4y^2}{64x^3y^6}$

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

$\frac{3x^4y^2}{64x^3y^6}=\frac{3}{64}x^{4-3}y^{2-6}=\frac{3}{64}xy^{-4}$

$\frac{3x}{64y^4}$

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Example Question #1 : Simplifying Exponents

Order the following from least to greatest:

$25^{100}$

$2^{300}$

$3^{500}$

$4^{400}$

$2^{600}$

Possible Answers:

$2^{300}, 2^{600}, 3^{500}, 25^{100}, 4^{400}$

$2^{600}, 2^{300}, 25^{100}, 3^{500}, 4^{400}$

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

$25^{100}, 2^{300}, 2^{600}, 3^{500}, 4^{400}$

$3^{500}, 4^{400}, 2^{300}, 25^{100}, 2^{600}$

Correct answer:

$2^{300}, 25^{100}, 2^{600}, 3^{500}, 4^{400}$

Explanation:

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent. Then they can be easily compared.

2^3^0^0 = (2^3)^1^0^0 = 8^1^0^0 , 3^5^0^0 = (3^5)^1^0^0 = 243^1^0^0 , 4^4^0^0 = (4^4)^1^0^0 = 256^1^0^0 , and 2^6^0^0 = (2^6)^1^0^0 = 64^1^0^0 .

Thus, ordering from least to greatest: 8^1^0^0, 25^1^0^0, 64^1^0^0, 243^1^0^0, 256^1^0^0 .

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Example Question #4 : Distributing Exponents (Power Rule)

Simplify:

$\frac{(x^{4})^{3}(5x)^{-5}}{xy}$

Possible Answers:

$\frac{125x^{2}}{y}$

$\frac{5x}{y}$

$\frac{x^{6}}{3125y}$

$\frac{1}{y}$

$\frac{x^{12}}{5y}$

Correct answer:

$\frac{x^{6}}{3125y}$

Explanation:

$\frac{(x^{4})^{3}(5x)^{-5}}{xy}$

Step 1: Distribute the exponents in the numberator.

$\frac{x^{12}5^{-5}x^{-5}}{xy}$

Step 2: Represent the negative exponents in the demoninator.

$\frac{x^{12}}{5^{5}x^{5}xy}$

Step 3: Simplify by combining terms.

$\frac{x^{12}}{3125x^{6}y}$

$\frac{x^6}{3125y}$

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Example Question #1 : Integer Exponents

Simplify:

(4x^3)^4

Possible Answers:

512x

16x^7

4x^7

$256x^{12}$

$4x^{12}$

Correct answer:

$256x^{12}$

Explanation:

Use the power rule to distribute the exponent:

$(4x^3)^4\rightarrow 4^4\cdot x^{3\cdot 4}\rightarrow 256x^{12}$

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Example Question #6 : Distributing Exponents (Power Rule)

Simplify:

$\left ( \frac{a^2b^{-4}}{a^3b^6c^{-7}} \right )^{-4}$

Possible Answers:

$\left ( \frac{a^4b^{40}}{c^{28}} \right )$

$\left ( \frac{a^{20}b^8}{c^{28}} \right )$

$\left ( \frac{ab}{c} \right )$

$\left ( \frac{c^{16}}{ab^4} \right )$

Correct answer:

$\left ( \frac{a^4b^{40}}{c^{28}} \right )$

Explanation:

Step 1: Distribute the exponent through the terms in parentheses:

$\left ( \frac{a^{-8}b^{16}}{a^{-12}b^{-24}c^{28}} \right )$

Step 2: Use the division of exponents rule. Subtract the exponents in the numerator from the exponents in the denominator:

${\color{Red} \left ( \frac{a^4b^{40}}{c^{28}} \right )}$

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Example Question #1 : Distributing Exponents (Power Rule)

Simplify.

$({x^{2a}}{x^{-b}})^2$

Possible Answers:

$x^{4a+b}$

$\frac{x^{4a}}{x^{2b}}$

$\frac{x^{2a}}{x^{2b}}$

$\frac{x^{4a}}{x^b}$

$x^{4a^2-4ab+b2}$

Correct answer:

$\frac{x^{4a}}{x^{2b}}$

Explanation:

When a power applies to an exponent, it acts as a multiplier, so 2a becomes 4a and -b becomes -2b. The negative exponent is moved to the denominator.

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Example Question #1 : Distributing Exponents (Power Rule)

Simplify (x^2y^4)^2 .

Possible Answers:

x^2y^2

x^4y^6

y^2

x^4y^8

xy^2

Correct answer:

x^4y^8

Explanation:

When faced with a problem that has an exponent raised to another exponent, the powers are multiplied: $x^{2\cdot2}y^{4\cdot2}$ then simplify: x^4y^8 .

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Example Question #1 : Distributing Exponents (Power Rule)

Solve: -25^0-(5^2)^0

Possible Answers:

Correct answer:

Explanation:

Solve each term separately. A number to the zeroth power is equal to 1, but be careful to apply the signs after the terms have been simplified.

-25^0-(5^2)^0 = -1-1 =-2

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Example Question #1 : Distributing Exponents (Power Rule)

Simplify this expression: $\left(8x^5x^3y^2\right)^2$

Possible Answers:

$16x^{34}y^4$

$64x^{16}y^4$

$64x^{30}y^4$

$64x^{34}y^4$

$16x^{12} y^4$

Correct answer:

$64x^{16}y^4$

Explanation:

$64x^{16}y^4$ is the correct answer because the order of operations were followed and the multiplication and power rules of exponents were obeyed. These rules are as follows: PEMDAS (parentheses,exponents, multiplication, division, addition, subtraction), for multiplication of exponents follow the format $a^m*a^n=a^{m+n}$ , and $(a^m)^n=a^{m*n}$ .

[8x^5x^3y^2]^2

First we simplify terms within the parenthesis because of the order of operations and the multiplication rule of exponents:

[8x^8y^2]^2

Next we use the power rule to distribute the outer power:

8^2(x^8)^2(y^2)^2 = $64x^{16}y^4$

**note that in the first step it isn't necessary to combine the two x powers because the individuals terms will still add to x^16 at the end if you use the power rule correctly. However, following the order of operations is a great way to avoid simple math errors and is relevant in many problems.

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Algebra II : Distributing Exponents (Power Rule)

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #2 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #4 : Distributing Exponents (Power Rule)

Example Question #1 : Integer Exponents

Example Question #6 : Distributing Exponents (Power Rule)

Example Question #1 : Distributing Exponents (Power Rule)

Example Question #1 : Distributing Exponents (Power Rule)

Example Question #1 : Distributing Exponents (Power Rule)

Example Question #1 : Distributing Exponents (Power Rule)

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