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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #288 : Simplifying Exponents

Solve: $8(2^{30})^3(4^{10})^3$

Possible Answers:

$4^{153}$

$2^{153}$

$8^{153}$

$2^{26}$

$4^{150}$

Correct answer:

$2^{153}$

Explanation:

In order to determine the value of this expression, change all the bases to base two.

$8(2^{30})^3(4^{10})^3=(2^3)(2^{30})^3(2^{2(10)})^3 =(2^3)(2^{30})^3(2^{20})^3$

Simplify the exponents separated by the inside and outside of the ending parentheses by multiplying the terms.

$(2^3)(2^{30})^3(2^{20})^3 = (2^3)\cdot (2^{30\times 3})\cdot (2^{20\times 3})$

Multiply the exponents.

$(2^3)\cdot (2^{30\times 3})\cdot (2^{20\times 3})= (2^3)\cdot (2^{90})\cdot (2^{60})$

Now that we are multiplying the similar bases with exponents, the exponents can be added.

$(2^3)\cdot (2^{90})\cdot (2^{60})=2^{3+90+60} = 2^{153}$

The answer is: $2^{153}$

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Example Question #291 : Simplifying Exponents

Evaluate: 2(2^2)^4

Possible Answers:

1024

$\textup{The answer is not given.}$

512

256

128

Correct answer:

512

Explanation:

In order to evaluate this expression, we can use the distributive property of exponents to simplify.

Multiply the powers together.

$2(2^2)^4 = 2(2)^{2(4)} = 2(2^8)$

This is also the same as: $2^1 \cdot 2^8 = 2^{1+8} = 2^9$

$2^9 = (2\cdot2\cdot2)^3 = 8^3 =512$

The answer is: 512

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Example Question #292 : Simplifying Exponents

Solve: $(3x^{45})^3$

Possible Answers:

$27x^{48}$

$3x^{135}$

$9x^{135}$

$9x^{48}$

$27x^{135}$

Correct answer:

$27x^{135}$

Explanation:

Solve by multiplying the exponents of each term inside the parentheses by the outer exponent of the parentheses.

The exponent of three inside the parentheses can be written as a one.

$(3x^{45})^3=(3^1x^{45})^3 = 3^3 x^{45\times 3} = 27x^{135}$

The answer is: $27x^{135}$

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Example Question #293 : Simplifying Exponents

Evaluate: $b(b^{15})^3$

Possible Answers:

$b^{19}$

$b^{45}$

$b^{90}$

$b^{46}$

$b^{18}$

Correct answer:

$b^{46}$

Explanation:

We will need to use the power rule of exponents to simplify the term enclosed in the parentheses.

$(a^b)^c = a^{bc}$

$b(b^{15})^3 = b\cdot b^{15\times3} = b\cdot b^{45}$

According to the addition rule of exponents, whenever common bases raised to a certain exponent are multiplied, their powers can be added.

$b\cdot b^{45}= b^1\cdot b^{45} = b^{1+45} = b^{46}$

The answer is: $b^{46}$

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Example Question #294 : Simplifying Exponents

Simplify: $(3^{30})^{25}$

Possible Answers:

$3^{750}$

$\textup{The answer is not given.}$

$3^{775}$

$3^{55}$

$3^{625}$

Correct answer:

$3^{750}$

Explanation:

The quantity of the inner term is raised to the power of 25, which mean that the inner term will be multiplied by itself 25 times.

This means that we can multiply the inside power with the outside power as the exponential property.

$(a^b)^c = a^{bc}$

$(3^{30})^{25} = 3^{30\times 25} = 3^{750}$

The answer is: $3^{750}$

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Example Question #291 : Simplifying Exponents

Solve: $4(2^6)^{\frac{1}{3}}$

Possible Answers:

$4\sqrt2$

$16\sqrt2$

Correct answer:

Explanation:

Use the product rule of exponents to simplify the exponents.

$(a^b)^c =a^{bc}$

Using this rule, we can rewrite the given problem.

$4(2^6)^{\frac{1}{3}} = 4(2^{6\times \frac{1}{3}}) =4(2^2)=4(4)=16$

The answer is:

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Example Question #292 : Simplifying Exponents

Evaluate: -(2^3)^2

Possible Answers:

Correct answer:

Explanation:

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

$(a^b)^c=a^{bc}$

Simplify the problem.

$-(2^3)^2= -(2^{3\times 2}) = -(2^6) = -64$

The answer is:

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Example Question #297 : Simplifying Exponents

Solve: $4(32^9)^{10}$

Possible Answers:

$8^{45}$

$32^{45}$

$2^{452}$

$4^{256}$

$2^{904}$

Correct answer:

$2^{452}$

Explanation:

We will need to rewrite this in base two.

$4(32^9)^{10} = 2^2[(2^5)^9]^{10} = 2^2\cdot [(2^5)^9]^{10}$

The exponents in the brackets and parentheses can be simplified by multiplication.

$2^2\cdot [(2^5)^9]^{10} = 2^2\cdot 2^{450}$

Add the exponents.

$2^2\cdot 2^{450} = 2^{2+450} = 2^{452}$

The answer is: $2^{452}$

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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 #291 : Simplifying Exponents

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

Possible Answers:

$6a^{11}$

$6a^{24}$

$2a^{11}$

$8a^{11}$

$8a^{24}$

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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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #288 : Simplifying Exponents

Example Question #291 : Simplifying Exponents

Example Question #292 : Simplifying Exponents

Example Question #293 : Simplifying Exponents

Example Question #294 : Simplifying Exponents

Example Question #291 : Simplifying Exponents

Example Question #292 : Simplifying Exponents

Example Question #297 : Simplifying Exponents

Example Question #111 : Distributing Exponents (Power Rule)

Example Question #291 : Simplifying Exponents

All Algebra II Resources

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