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Pre-Algebra : Power Rule of Exponents

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Example Questions

Example Question #1 : Power Rule Of Exponents

Simplify:

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

Possible Answers:

$27x^{6}y^{3}$

$3x^{2}y^{3}$

$9x^{2}y$

$9x^{6}y^{3}$

$3x^{5}y^{3}$

Correct answer:

$27x^{6}y^{3}$

Explanation:

Recall the Power Rule of Exponents: $\left ( x^{m} \right )^{n}=x^{m*n}$ .

To simplify the expression $\left ( 3x^{2}y \right )^{3}$ , use this rule to distribute the exponents:

$\left ( 3x^{2}y \right )^{3}=3^{3}x^{2*3}y^{3}=27x^{6}y^{3}$

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Example Question #2 : Power Rule Of Exponents

Simplify:

$x^{5}\left ( x^{2}y \right )^{2}$

Possible Answers:

$x^{9}y^{2}$

$x^{12}y^{2}$

$x^{10}y^{2}$

$x^{20}y^{2}$

$x^{7}y^{2}$

Correct answer:

$x^{9}y^{2}$

Explanation:

Recall the following rules:

The Product Rule of Exponents says: $x^{m}*x^{n}=x^{m+n}$

The Power Rule of Exponents says: $\left ( x^{m} \right )^{n}=x^{m*n}$ and $\left ( x^{m}y \right )^{n}=x^{m*n}y^{n}$

Step 1: Use the Power Rule of Exponents to distribute the exponents:

$\left ( x^{2}y \right )^{2}=x^{2*2}y^{2}=x^{4}y^{2}$

Step 2: Use the Product Rule of Exponents to simplify further:

$x^{5}x^{4}y^{2}=x^{5+4}y^{2}=x^{9}y^{2}$

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Example Question #3 : Power Rule Of Exponents

Simplify.

$\frac{x^{8}}{x^{6}}$

Possible Answers:

$x^{14}$

$x^{2}$

$\frac{1}{x^{2}}$

$x^{-2}$

Correct answer:

$x^{2}$

Explanation:

The Quotient of Powers Property states when you divide two powers with the same base, you subtract the exponents.

In this case, the exponents are 8 and 6:

$x^{8-6}=x^{2}$

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Example Question #1 : Power Rule Of Exponents

Simplify:

$\small \small (4xy^3z^2)^2$

Possible Answers:

$\small 16x^3y^5z^4$

$\small \small 16x^2y^6z^4$

$\small 8x^2y^6z^4$

$\small \small 8x^3y^5z^4$

Correct answer:

$\small \small 16x^2y^6z^4$

Explanation:

Recall, to distribute exponents through a polynomial, you must multiply the exponents. Thus, our expression can be simplified as follows:

$\small \small (4xy^3z^2)^2$

$\small 4^{1*2}x^{1*2}y^{3*2}z^{2*2}$

$\small 4^2x^2y^6z^4$

$\small 16x^2y^6z^4$

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Example Question #2 : Power Rule Of Exponents

Simplify:

x^4y(x^3y^2)^2

Possible Answers:

$x^{10}y^5$

$x^{7}y^5$

$x^{24}y^4$

$x^{9}y^4$

$4x^{10}y^5$

Correct answer:

$x^{10}y^5$

Explanation:

Recall the following rules:

The Product Rule of Exponents says: $x^m * x^n = x^{m+n}$

The Power Rule of Exponents says: $(x^m)^n = x^{m+n}$ and $(x^my)^n = x^{m*n}y^n$

Step 1: Use the Power Rule of Exponents to distribute the exponents:

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

Step 2: Use the Product Rule of Exponents to simplify further:

$x^4y(x^6y^4) = x^{4+6}y^{1+4} = x^{10}y^5$

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Example Question #3 : Power Rule Of Exponents

Simplify:

$(5x^{2}y^{^{3}})^{2}$

Possible Answers:

$10x^{4}y^{6}$

$25x^{2}y^{6}$

$125x^{4}y^{6}$

$25x^{4}y^{6}$

$25x^{4}y^{5}$

Correct answer:

$25x^{4}y^{6}$

Explanation:

Recall the Power Rule of Exponents: $(x^{m})^{n}= x^{m*n}$

To simplify the expression $(5x^{2}y^{^{3}})^{2}$ , use this rule to distribute the exponents:

$(5x^{2}y^{^{3}})^{2} = 5^{2}x^{2*2}y^{3*2} = 25x^{4}y^{6}$

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Example Question #4 : Power Rule Of Exponents

y(2x^2y^3)^2

Possible Answers:

2x^4y^7

4x^4y^7

4x^4y^6

4x^4y^8

4x^4y^5

Correct answer:

4x^4y^7

Explanation:

First, distribute the exponent outside of the parentheses to each of the elements inside of the parentheses, including the 2.

(2x^2y^3)^2 = 2^2x^4y^6

Remember that in this case, when an exponent is raised to another power, the exponents multiply.

Now we need to mutiply that answer by the outside .

4x^4y^6*y = 4x^4y^7

For this last step, remember that the exponents on the add.

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Example Question #1 : Power Rule Of Exponents

What is the simplified version of the following polynomial?

$\small x^{24}* x^2 + 2^2 + 2^2$

Possible Answers:

$\small \small x^{48}+8$

$\small \small x^{26}+16$

$\small x^{26}+8$

$\small 8\small x^{26}$

$\small \small 8x^{48}$

Correct answer:

$\small x^{26}+8$

Explanation:

The power rule of an exponent means that when two exponential expressions with the same base are multiplied, you add the exponents together. In this case,

$\small \small \small \small x^{24}*x^2 = x^{26}$

$\small x^{24}$ can be thought of as a "string" of 24 $\small x$ variables being multiplied together, so by multiplying that string by another 2 variable units, you have seamlessly extended the chain by two units.

For $\small 2^2 +2^2$ , it is important to remember that the power rule does no apply during addition, so the exponent is applied first, then the exponential values are added following the order of operation, so the final expression is:

$\small \small x^{24 +2} + 4+4$

or $\small x^{26} + 8$

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Example Question #2 : Power Rule Of Exponents

Simplify:
(x^5)(x^8)

Possible Answers:

$x^{40}$

$x^{13}$

$x^{\frac{5}{8}}$

x^3

Correct answer:

$x^{13}$

Explanation:

The product rule of exponents states that when you're multiplying two powers with the same base, you add the exponents:

$(x^a)(x^b)=x^{a+b}$

In this instance,

$(x^5)(x^8)=x^{5+8}=x^{13}$

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Example Question #1 : Power Rule Of Exponents

Simplify:
$\small (x^3)^4$

Possible Answers:

$\small \small x^7$

$\small x^1$

$\small x^{12}$

$\small x^{\frac{3}{4}}$

Correct answer:

$\small x^{12}$

Explanation:

The "power rule" states that when you raise a power to a power, you mutiploy the exponents:
$\small (x^a)^b=x^{a\times b}$

$\small (x^3)^4=x^{3\times4}=x^{12}$

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Pre-Algebra : Power Rule of Exponents

Study concepts, example questions & explanations for Pre-Algebra

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