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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 #12 : Distributing Exponents (Power Rule)

Solve: $2(27)^{\frac{4}{3}}$

Possible Answers:

$81\sqrt{2}$

162

$162\sqrt[3]{2}$

Correct answer:

162

Explanation:

First convert into a known base. The number can be rewritten as 3^3 .

Rewrite the expression.

$2(27)^{\frac{4}{3}}=2(3^3)^{\frac{4}{3}}$

Use the power rule to multiply the exponents.

$2(3^3)^{\frac{4}{3}} = 2(3)^4$

Use order of operations to evaluate the expression.

$2(3)^4 = 2(3\cdot 3 \cdot 3\cdot 3) = 2(81)= 162$

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Example Question #1 : Expressions & Equations

Which of the following is equivalent to the expression (7x^5)^3 ?

Possible Answers:

$343x^{8}$

$49x^{2}$

$7x^{15}$

$343x^{15}$

Correct answer:

$343x^{15}$

Explanation:

Which of the following is equivalent to the expression (7x^5)^3 ?

We can rewrite the given expression by distributing the exponent on the outside.

(7x^5)^3=7^3(x^5)^3

Now, this may look a little messier, but we need to recall that when we distribute an exponent through parentheses as we are trying to do above, we need to multiple the exponent on the inside by the number on the outside.

In a general sense it looks like this:

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

For our specific problem, it looks like this:

$7^3(x^5)^3=7^3x^{15}=343x^{15}$

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Example Question #11 : Distributing Exponents (Power Rule)

Simplify: $x^{\frac{5}{4}}$

Possible Answers:

$x\sqrt[4]{x}$

$x\sqrt[5]{x}$

$\sqrt[4]{x}$

x^5

Correct answer:

$x\sqrt[4]{x}$

Explanation:

To simplify this expression, recall that it could be written as this: $\sqrt[4]{x^5}$ . Since we're dealing with a quartic root, for every 4 of the same term, cross them out underneath the radical and bring one of those terms out. Therefore, we are only left with one x underneath the radical. The answer is: $x\sqrt[4]{x}$ .

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Example Question #1 : Integer Exponents

Simplify the following expression:

(9x^6y^2z^5)^2

Possible Answers:

$81x^{12}y^4z^{10}$

$81x^{8}y^4z^{7}$

$18x^{12}y^4z^{10}$

$11x^{12}y^4z^{10}$

Correct answer:

$81x^{12}y^4z^{10}$

Explanation:

Simplify the following expression:

(9x^6y^2z^5)^2

To raise exponents to another power, we need to multiply them:

$(9x^6y^2z^5)^2=9^2x^{6*2}y^{2*2}z^{5*2}=81x^{12}y^4z^{10}$

So we get:

$81x^{12}y^4z^{10}$

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Example Question #14 : Distributing Exponents (Power Rule)

Simplify the following expression:

(5x^4t^6h^2)^5

Possible Answers:

$625x^{20}t^{30}h^{10}$

$3125x^{20}t^{30}h^{10}$

$3125x^{9}t^{11}h^{7}$

$5x^{20}t^{30}h^{10}$

Correct answer:

$3125x^{20}t^{30}h^{10}$

Explanation:

Simplify the following expression:

(5x^4t^6h^2)^5

To simplify exponents that are being raised to a higher power, we need to multiply each term's exponent by the exponent outside of the parentheses.

$(5x^4t^6h^2)^5=5^5x^{4*5}t^{6*5}h^{2*5}$

Follow up by actually multiplying through

$5^5x^{4*5}t^{6*5}h^{2*5}=3125x^{20}t^{30}h^{10}$

Making our answer:

$3125x^{20}t^{30}h^{10}$

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Example Question #15 : Distributing Exponents (Power Rule)

Simplify the following expression by distributing the exponent:

$(3y^4t^8x^{12})^5$

Possible Answers:

$3y^{20}t^{40}$

$9y^{9}t^{13}x^{17}$

$9y^{20}t^{40}x^{60}$

$243y^{20}t^{40}x^{60}$

Correct answer:

$243y^{20}t^{40}x^{60}$

Explanation:

Simplify the following expression by distributing the exponent:

$(3y^4t^8x^{12})^5$

When we are distributing an exponent, we want to multiple the exponent of each term within the parentheses by the exponent outside the parentheses.

$3^{1*5}y^{4*5}t^{8*5}x^{12*5}$

Finally, perform the multiplication to get your answer:

$243y^{20}t^{40}x^{60}$

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Example Question #1 : Expressions & Equations

Simplify.

$(x^4)^{6}$

Possible Answers:

$x^{10}$

$x^{24}$

$x^{-2}$

$x^{46}$

x^2

Correct answer:

$x^{24}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(x^4)^{6}=x^{4*6}=x^{24}$

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Example Question #2 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Simplify.

$(x^{-3})^{-4}$

Possible Answers:

$x^{-12}$

$x^{12}$

$x^{7}$

$x^{-7}$

$x^{34}$

Correct answer:

$x^{12}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(x^{-3})^{-4}=x^{-3*-4}=x^{12}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify.

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

Possible Answers:

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

$\frac{x^{15}}{y^{20}}$

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

$\frac{x^{8}}{y^{9}}$

$\frac{x^8}{y^4}$

Correct answer:

$\frac{x^{15}}{y^{20}}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(\frac{x^3}{y^4})^{5}=\frac{x^{3*5}}{y^{4*5}}=\frac{x^{15}}{y^{20}}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify.

$(\frac{x^5}{y^3})^3$

Possible Answers:

$\frac{x^{15}}{y^{13}}$

$\frac{x^{15}}{y^8}$

$\frac{x^{15}}{y^9}$

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

$\frac{x^{15}}{y^3}$

Correct answer:

$\frac{x^{15}}{y^9}$

Explanation:

When an exponent is being raised by another exponent, we just multiply the powers of the exponents and keep the base the same.

$(\frac{x^5}{y^3})^{3}=\frac{x^{5*3}}{y^{3*3}}=\frac{x^{15}}{y^{9}}$

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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 #12 : Distributing Exponents (Power Rule)

Example Question #1 : Expressions & Equations

Example Question #11 : Distributing Exponents (Power Rule)

Example Question #1 : Integer Exponents

Example Question #14 : Distributing Exponents (Power Rule)

Example Question #15 : Distributing Exponents (Power Rule)

Example Question #1 : Expressions & Equations

Example Question #2 : Generate Equivalent Numerical Expressions: Ccss.Math.Content.8.Ee.A.1

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

All Algebra II Resources

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