Negative Exponents - Algebra II

Card 0 of 472

Question

Simplify:

$10^{-5}$

Answer

When an exponent is negative, we express as such:

$x^{-a}=\frac{1}{x^a}$

is the positive exponent, and is the base.

$10^{-5}=\frac{1}{10^5}= \frac{1}{100000}$ .

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Question

Simplify:

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

Answer

When an exponent is negative, we express as such:

$x^{-a}=\frac{1}{x^a}$

is the positive exponent, and is the base.

$\frac{1}{2}^{-2}=\frac{1}{\frac{1}{2^2}}=\frac{1}{\frac{1}{4}}=4$

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Question

Simplify:

$(-2)^{-2}$

Answer

When an exponent is negative, we express as such:

$x^{-a}=\frac{1}{x^a}$

is the positive exponent, and is the base.

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

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Question

Simplify:

$-3^{-3}$

Answer

When an exponent is negative, we express as such:

$x^{-a}=\frac{1}{x^a}$

is the positive exponent, and is the base.

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

Remember the negative sign is not part of the base value.

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Question

Evaluate:

$15^{-4}$

Answer

In order to convert negative exponents, we will use the following formula:

$x^{-a}=\frac{1}{x^a}$

In this formula, is the positive exponent and is the base.

$15^{-4}=\frac{1}{15^4}$

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Question

Evaluate:

$(\frac{1}{6})^{-3}$

Answer

In order to convert negative exponents, we will use the following formula:

$x^{-a}=\frac{1}{x^a}$

In this formula, is the positive exponent and is the base.

$6^{-3}=\frac{1}{(\frac{1}{6})^3}=\frac{1}{\frac{1}{216}}=216$

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Question

Evaluate:

$18^{-1}$

Answer

When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$ .

In this format, represents the base and represents the exponent. A negative exponent becomes positive when it is rewritten as a reciprocal.

Therefore:

$18^{-1}=\frac{1}{18^1}=\frac{1}{18}$

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Question

Evaluate:

$24^{-2}$

Answer

When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$ .

In this format, represents the base and represents the exponent. A negative exponent becomes positive when it is rewritten as a reciprocal.

Therefore:

$24^{-2}=\frac{1}{24^2}=\frac{1}{576}$

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Question

Evaluate:

$\left(\frac{1}{7}\right)^{-3}$

Answer

When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$ .

In this format, represents the base and represents the exponent. A negative exponent becomes positive when it is rewritten as a reciprocal.

Therefore:

$\left(\frac{1}{7}\right)^{-3}=\frac{1}{\frac{1}{7^3}}=\frac{1}{\frac{1}{343}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{1}{\frac{1}{343}}=1 \times \frac{343}{1}= 343$

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Question

Evaluate:

$\left(\frac{2}{3}\right)^{-4}$

Answer

When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$ .

In this format, represents the base and represents the exponent. A negative exponent becomes positive when it is rewritten as a reciprocal.

Therefore:

$\left(\frac{2}{3}\right)^{-4}=\frac{1}{(\frac{2}{3})^3}=\frac{1}{\frac{16}{81}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{1}{\frac{16}{81}}=1\times\frac{81}{16}=\frac{81}{16}$

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Question

Evaluate: $5^{-8}$

Answer

When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$5^{-8}=\frac{1}{5^8}$

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Question

Simplify: $12^{-12}$

Answer

When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$12^{-12}=\frac{1}{12^{12}}$

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Question

Simplify: $\frac{1}{64}^{-3}$

Answer

When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$\frac{1}{64}^{-3}=\frac{1}{\frac{1}{64^3}}=\frac{1}{\frac{1}{262144}}=262144$

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Question

Evaluate: $\frac{1}{64}^{-\frac{1}{3}}$

Answer

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base . When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$\frac{1}{64}^{-\frac{1}{3}}=\frac{1}{\frac{1}{\sqrt[3]{64}}}=\frac{1}{\frac{1}{4}}=4$

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Question

Evaluate: $\frac{1}{8}^{-\frac{2}{3}}$

Answer

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base . When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$\frac{1}{8}^{-\frac{2}{3}}=\frac{1}{\frac{1}{\sqrt[3]{8^2}}}=\frac{1}{\frac{1}{4}}=4$

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Question

Evaluate $21^{-2}$

Answer

When expressing negative exponents, we rewrite as such:

$x^{-a}=\frac{1}{x^a}$

in which is the positive exponent raising base .

$21^{-2}=\frac{1}{21^2}=\frac{1}{441}$

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Question

Evaluate $12^{-\frac{1}{2}}$

Answer

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

When expressing negative exponents, we rewrite as such:

$x^{-a}=\frac{1}{x^a}$

in which is the positive exponent raising base .

$12^{-\frac{1}{2}}=\frac{1}{(\sqrt{12})}=\frac{1}{2{\sqrt{3}}}=\frac{\sqrt{3}}{6}$

Remember when getting rid of radicals, just multiply top and bottom by that radical.

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Question

Evaluate $-4^{-4}$

Answer

When expressing negative exponents, we rewrite as such:

$x^{-a}=\frac{1}{x^a}$

in which is the positive exponent raising base .

$-4^{-4}=-\frac{1}{4^4}=-\frac{1}{256}$

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Question

Evaluate $(-5)^{-4}$

Answer

When expressing negative exponents, we rewrite as such:

$x^{-a}=\frac{1}{x^a}$

in which is the positive exponent raising base .

$(-5)^{-4}=\frac{1}{(-5)^4}=\frac{1}{625}$

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Question

Evaluate $\frac{1}{2}^{-3}$

Answer

When expressing negative exponents, we rewrite as such:

$x^{-a}=\frac{1}{x^a}$

in which is the positive exponent raising base .

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

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