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Algebra II : Distributing Exponents (Power Rule)

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Example Question #111 : Distributing Exponents (Power Rule)

Simplify: $(3x^4)^{3}$

Possible Answers:

$9x^{7}$

$27x^{7}$

$3x^{12}$

$9x^{12}$

$27x^{12}$

Correct answer:

$27x^{12}$

Explanation:

To simplify this expression, we will need to multiply the exponents of the inside quantity by the outer exponent.

$(3x^4)^{3} = (3^{1}\cdot x^4)^{3} = (3^{1\cdot 3 }\cdot x^{4\cdot 3})$

Simplify the terms by multiplication.

$3^3 \cdot x^{12} = 27x^{12}$

The answer is: $27x^{12}$

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Example Question #111 : Distributing Exponents (Power Rule)

Solve: $(2a^8)^{3}$

Possible Answers:

$2a^{11}$

$6a^{24}$

$8a^{11}$

$6a^{11}$

$8a^{24}$

Correct answer:

$8a^{24}$

Explanation:

In order to solve this expression, use the power rule of exponents.

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

Notice that there is also a two in front of the variable. We will need to convert that as an exponent as well.

$(2a^8)^{3}=(2^1\cdot a^8)^{3}= 2^3\cdot a^{8\times 3}$

Simplify the expression.

The answer is: $8a^{24}$

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Example Question #113 : Distributing Exponents (Power Rule)

Evaluate: $2(4^{10})(8^{20})^3$

Possible Answers:

$2^{1801}$

$2^{70}$

$2^{71}$

$2^{201}$

$2^{261}$

Correct answer:

$2^{201}$

Explanation:

To evaluate this expression, we will need to convert to common bases in order to use the power rule for exponents.

4=2^2

8=2^3

Rewrite the problem.

$2(4^{10})(8^{20})^3 =2(2^{2(10)})(2^{3(20)})^3= 2\cdot 2^{20}\cdot(2^{60})^3$

Do not mix the additive rule of exponents with the multiplicative rule.

$a^b\cdot a^c = a^{b+c}$

$(x^y)^z = x^{yz}$

Use these rules to simplify the terms.

$2\cdot 2^{20}\cdot(2^{60})^3 = 2^1\cdot 2^{20}\cdot2^{180} = 2^{1+20+180} =2^{201}$

The answer is: $2^{201}$

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Example Question #114 : Distributing Exponents (Power Rule)

Solve: $(8^{4})^{15}$

Possible Answers:

$8^{19}$

$2^{170}$

$4^{80}$

$2^{60}$

$8^{60}$

Correct answer:

$8^{60}$

Explanation:

Use the power rule of exponents to simplify this expression.

$(x^{Y})^Z = x^{YZ}$

Follow this rule to simplify the exponents.

$(8^{4})^{15} = 8^{4\times 15} = 8^{60}$

The answer is: $8^{60}$

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Example Question #115 : Distributing Exponents (Power Rule)

Simplify: $[(2^{-3})^{-2}]^{-\frac{1}{2}}$

Possible Answers:

$\frac{1}{64}$

$\frac{1}{8}$

$\frac{1}{32}$

Correct answer:

$\frac{1}{8}$

Explanation:

We can use the power rule of exponents to simplify this expression.

$(x^{Y})^Z =x^{YZ}$

$[(2^{-3})^{-2}]^{-\frac{1}{2}} = 2^{(-3)\cdot (-2 )\cdot( -\frac{1}{2})}$

The expression becomes:

$2^{-3}$

We can rewrite this as a fraction.

$2^{-3} = \frac{1}{2^3}=\frac{1}{8}$

The answer is: $\frac{1}{8}$

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Example Question #116 : Distributing Exponents (Power Rule)

Simplify: $b(b^{-3})^{-10}$

Possible Answers:

$\frac{1}{b^{14}}$

$b^{14}$

$b^{31}$

$b^{29}$

$\frac{1}{b^{12}}$

Correct answer:

$b^{31}$

Explanation:

Use the product rule of exponents to simplify the terms in parentheses.

$(b^{X})^Y= b^{XY}$

$b(b^{-3})^{-10} = b(b^{-3\times -10}) = b(b^{30})$

This expression is the same as: $b^1 \cdot b^{30}$

The answer is: $b^{31}$

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Example Question #117 : Distributing Exponents (Power Rule)

Solve: (5x^3)^4

Possible Answers:

$5x^{7}$

$625x^{12}$

$625x^{7}$

$5x^{12}$

$20x^{12}$

Correct answer:

$625x^{12}$

Explanation:

In order to solve this expression, we will need to use the power rule of exponents.

$(a^{x})^y = a^{xy}$

We can distribute the outer exponent to the exponents inside the parentheses. Do not ignore the five, which equals 5^1 .

$(5x^3)^4 = (5^{1\times 4}\cdot x^{3\times 4}) = 5^4 \cdot x^{12}$

The answer is: $625x^{12}$

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Example Question #301 : Simplifying Exponents

Solve: $\frac{1}{2}(64^{5})^{10}$

Possible Answers:

$4^{99}$

$2^{299}$

$4^{25}$

$2^{49}$

$4^{13}$

Correct answer:

$2^{299}$

Explanation:

In order to solve this, we can change this expression to base two.

The fraction can be rewritten as a negative exponent.

$\frac{1}{2} =2^{-1}$

The inner term in the parentheses is equal to:

64= 2^6

Rewrite the expression.

$\frac{1}{2}(64^{5})^{10} = 2^{-1} (2^{6(5)})^{(10)}$

Using the product rule of exponents, we can multiply all the powers together.

$2^{-1} (2^{6\times 5\times 10})= 2^{-1} (2^{300}) = 2^{-1} \cdot (2^{300})$

Now that we are multiplying two common bases with exponents, we can add the powers.

$2^{-1+300} = 2^{299}$

The answer is: $2^{299}$

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Example Question #119 : Distributing Exponents (Power Rule)

Solve: 5[(5^3)^4]^5

Possible Answers:

$5^{18}$

$5^{61}$

$5^{13}$

$5^{12}$

$5^{60}$

Correct answer:

$5^{61}$

Explanation:

The exponents that are surrounded by the brackets and parentheses can be multiplied based on the product rule for exponents.

$5[(5^3)^4]^5 = 5^1\cdot [5^{3\times 4\times 5}] = 5^1\cdot 5^{60}$

Now that we have bases similar, we can use the additive rule of exponents to combine both terms.

The answer is: $5^{61}$

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Example Question #120 : Distributing Exponents (Power Rule)

Solve: $[(3^{-3})^2]^{\frac{1}{2}}$

Possible Answers:

$\frac{1}{27}$

$\frac{1}{9}$

Correct answer:

$\frac{1}{27}$

Explanation:

According to the product rule of exponents, we can multiply the two powers together in the parentheses.

$(x^B)^C = x^{B\times C}$

This means that:

$(3^{-3})^2 = 3^{-6}$

Replace this term in the bracket.

$[(3^{-3})^2]^{\frac{1}{2}} = [3^{-6}]^{\frac{1}{2}}$

Repeat the process for these exponents.

$[3^{-6}]^{\frac{1}{2}} = 3^{-6\times \frac{1}{2}} = 3^{-3}$

This negative exponent can be simplified as a fraction.

$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$

The answer is: $\frac{1}{27}$

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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 #111 : Distributing Exponents (Power Rule)

Example Question #111 : Distributing Exponents (Power Rule)

Example Question #113 : Distributing Exponents (Power Rule)

Example Question #114 : Distributing Exponents (Power Rule)

Example Question #115 : Distributing Exponents (Power Rule)

Example Question #116 : Distributing Exponents (Power Rule)

Example Question #117 : Distributing Exponents (Power Rule)

Example Question #301 : Simplifying Exponents

Example Question #119 : Distributing Exponents (Power Rule)

Example Question #120 : Distributing Exponents (Power Rule)

All Algebra II Resources

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