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High School Math : Distributing Exponents

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Example Questions

Example Question #1 : Simplifying Exponents

$\log_{x}27e^{3}=3$ $\displaystyle \log_{x}27e^{3}=3$

Solve for $\displaystyle x$.

Possible Answers:

$\displaystyle 4e$

1/2e $\displaystyle 1/2e$

$\displaystyle e$

$\displaystyle 2e$

$\displaystyle 3e$

Correct answer:

$\displaystyle 3e$

Explanation:

First, set up the equation: x^3=27 $\displaystyle x^3=27$ e^3 $\displaystyle e^3$. Simplifying this result gives x = 3 $\displaystyle x = 3$$\displaystyle e$.

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Example Question #1 : Simplifying Exponents

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

Possible Answers:

Correct answer:

Explanation:

$\displaystyle 16^4 = (2^4)^4 = 2^1^6$. Thus, $\displaystyle z$ is equal to 16.

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Example Question #2 : Simplifying Exponents

Order the following from least to greatest:

$25^{100}$ $\displaystyle 25^{100}$

$2^{300}$ $\displaystyle 2^{300}$

$3^{500}$ $\displaystyle 3^{500}$

$4^{400}$ $\displaystyle 4^{400}$

$2^{600}$ $\displaystyle 2^{600}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

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.

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

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

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Example Question #1 : Review And Other Topics

Simplify the expression:

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

Possible Answers:

Cannot be simplified

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

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

$\displaystyle 3xy$

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

Correct answer:

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

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:

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

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

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

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

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

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

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High School Math : Distributing Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #2 : Simplifying Exponents

Example Question #1 : Review And Other Topics

All High School Math Resources

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