Exponents - Algebra II

Card 0 of 3072

Question

Simplify:

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$23^7*23^8=23^{7+8}=23^{15}$

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Question

Simplify: $2^4*2^{-12}$

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$2^4*2^{-12}=2^{4+(-12)}=2^{-8}$

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Question

Simplify: $(-3)^7*(-3)^{-2}$

Answer

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$(-3)^7*(-3)^{-2}=(-3)^{7+(-2)}=(-3)^{5}$

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Question

Simplify:

Answer

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^5*x^6=x^{5+6}=x^{11}$

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Question

Simplify: $x^{12}*x^{-8}$

Answer

When multiplying exponents with the same base, we just keep the base the same and add the exponents.

$x^{12}*x^{-8}=x^{12+(-8)}=x^4$

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Question

Simplify:

$(5^{-8})^{-12}$

Answer

When dealing with exponents being raised by another exponent, we just multiply the powers and keep the base the same.

$(5^{-8})^{-12}=5^{-8*-12}=5^{96}$

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Question

Simplify: $(3a^2b^{-3}c^4)^2$

Answer

To simplify this expression, square every term in the parentheses:

$3^2a^4b^{-6}c^8$ .

Then simplify and get rid of the negative exponent by putting the b term on the denominator:

$\frac{9a^4c^8}{b^6}$ .

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Question

Simplify:

Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(4^7)^7=4^{7*7}=4^{49}$

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Question

What is $5^{-2}$ the same as?

Answer

While a positive exponent says how many times to multiply by a number, a negative exponent says how many times to divide by the number.

To solve for negative exponents, just calculate the reciprocal.

$5^{-2}=\frac{1}{5^{2}}=\frac{1}{25}$

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Question

Solve: $32^{-\frac{1}{5}}$

Answer

To evaluate a negative exponent, convert the exponent to positive by taking the inverse.

$32^{-\frac{1}{5}}= \frac{1}{32^{\frac{1}{5}}}=\frac{1}{2}$

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Question

Simplify:

$(8^7)^{-2}$

Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(8^7)^{-2}=8^{7*-2}=8^{-14}$

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Question

Simplify:

$(5^{-5})^{-5}$

Answer

When an exponent is raised by another exponent, we will multiply the exponents and keep the base the same.

Simplify:

$(5^{-5})^{-5}=5^{-5*-5}=5^{25}$

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Question

Convert the exponent to radical notation.

$x^{\frac{3}{7}}$

Answer

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$

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Question

Simplify:

$[x^{1/2}]^{7/3}$

Answer

$[x^{m}]^{n}=x^{m\times n}$

$x^{1/2\times 7/3}=x^{7/6}$

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Question

Simplify the expression:

$\small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$

Answer

Remember that fraction exponents are the same as radicals.

$\small 16^{\frac{1}{2}}=\sqrt{16}=4$

$256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

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

$\small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$

Either method, we then need to multiply to two terms.

$\small (4)(64)=256$

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Question

Write the product of $\small a^\frac{3}{4}a^\frac{3}{8}a^\frac{5}{2}$ in radical form

Answer

This problem relies on the key knowledge that $\small x^\frac{a}{b}=\sqrt[b]{x^a}$ and that the multiplying terms with exponents requires adding the exponents. Therefore, we can rewrite the expression thusly:

$\small a^\frac{3}{4}a^\frac{3}{8}a^\frac{5}{2}=a^{\frac{3}{4}+\frac{3}{8}+\frac{5}{2}}=a^{\frac{29}{8}}=\sqrt[8]{a^{29}}$

Therefore, $\small \sqrt[8]{a^{29}}$ is our final answer.

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Question

Evaluate the following expression:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}$

Answer

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}=\frac{3}{5}$

or

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}= \sqrt[3]{\frac{27}{125}}=\frac{\sqrt[3]{27}}{\sqrt[3]{125}}=\frac{3}{5}$

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Question

Simplify:

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}$

Answer

Keep in mind that when you are dividing exponents with the same base, you will want to subtract the exponent found in the denominator from the exponent found in the numerator.

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

$\frac{1}{2}-(-\frac{3}{2})=2$

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

Since the exponent is negative, you will want to put the in the denominator in order to make it positive.

So then,

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}=\frac{x^2}{y^2}$

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Question

Find the value of $9^{\frac{3}{2}}$ .

Answer

When you have a number or value with a fractional exponent,

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

or

$x^{\frac{a}{b}}=(\sqrt[b]{x})^a$

So then,

$9^{\frac{3}{2}}=\sqrt[2]{9^3}=\sqrt{729}=27$

$9^{\frac{3}{2}}=(\sqrt9)^3=3^3=27$

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Question

Find the value of $64^{\frac{4}{3}}$

Answer

When you have a number or value with a fractional exponent,

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

or

$x^{\frac{a}{b}}=(\sqrt[b]{x})^a$

So then,

$64^{\frac{4}{3}}=\sqrt[3]{64^4}=\sqrt[3]{2^{24}}=2^8=256$

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