Negative Exponents - Algebra II

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Question

Solve for :

$(x+5)^{-3} = -1$

Answer

Raise both sides of the equation to the inverse power of to cancel the exponent on the left hand side of the equation.

$\rightarrow ((x+5)^{-3})^{-\frac{1}{3}} = (-1)^{-\frac{1}{3}}$

$\rightarrow x+5 = -1$

Subtract from both sides:

$\rightarrow (x+5) - 5 = (-1)-5$

$\rightarrow x = -6$

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Question

Simplify the following expression

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

Answer

$\left (\frac{f^{-3}}{g^{-2}} \right )^{4}=\left (\frac{g^{2}}{f^{3}} \right )^{4}=\frac{g^{2\times 4}}{f^{3\times4}}=\frac{g^{8}}{f^{12}}$

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Question

Simplify the following expression:

$\frac{4x^{-2}y^{3}}{18x^{-4}y^{2}}$

Answer

$\frac{4x^{-2}y^3}{18x^{-4}y^{2}}=\frac{4x^{4}y^{3}}{18x^{2}y^{2}}=\left (\frac{4}{18} \right )x^{4-2}y^{3-2}=\left (\frac{4}{18} \right )x^{2}y=\frac{2x^{2}y}{9}$

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Question

Simplify the following expression

$\frac{(a^{2})^{-7}b^{0}}{a^{3}b^{-4}}$

Answer

$\frac{(a^{2})^{-7}b^{0}}{a^{3}b^{-4}}=\frac{a^{2\times \left(-7\right)}b^0}{a^{3}b^{-4}}=\frac{a^{-14}b^{0}}{a^{3}b^{-4}}=a^{-14-3}b^{0-(-4)}=a^{-17}b^{4}=\frac{b^{4}}{a^{17}}$

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Question

Simplify the following expression

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

Answer

$\left (\frac{m}{n} \right )^{-2}\left (\frac{n}{m} \right )^{4}=\left (\frac{n}{m} \right )^{2}\left (\frac{n}{m} \right )^{4}=\frac{n^{2+4}}{m^{2+4}}=\frac{n^{6}}{m^{6}}$

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Question

Simplify the following expression

$\left (\frac{34x}{2y} \right )(17xy)^{-1}$

Answer

$\left (\frac{34x}{2y} \right )(17xy)^{-1}=\left (\frac{34x}{2y} \right )\left (\frac{1}{17xy} \right )=\frac{34x}{34xy^2}=\frac{1}{y^2}$

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Question

Represent the fraction using only positive exponents:

$\frac{x^{-12}}{x^{-4}y^{-8}}$

Answer

$\frac{x^{-12}}{x^{-4}y^{-8}}$

Negative exponents are the reciprocal of their positive counterpart. For example:

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

Therefore:

$\frac{x^{-12}}{x^{-4}y^{-8}} = \frac{x^4y^8}{x^{12}}$

This simplifies to:

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

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Question

Solve the equation for n:

$2^{n+4}=\frac{1}{64}$

Answer

Rewrite the right-hand-side so that each side has the same base:

$2^{n+4}=2^{-6}$

Use the Property of Equality for Exponential Functions:

Solving for :

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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:

$\frac{x^4y^{-3}z^{10}}{x^{-9}y^{11}z^{-3}}$

Answer

To simplify this expression, first make all of the negative exponents positive. That means putting them in the opposite position (if they're in the numerator, put it in the denominator and vice versa).

It should then look like:

$\frac{x^4\cdot x^9\cdot z^{10}\cdot z^3}{y^{14}}$ .

Then, combine like terms. Remember, if bases are the same, add exponents!

Therefore, your answer is:

$\frac{x^{13}z^{13}}{y^{14}}$

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Question

Simplify the expression: $\frac{6a^3b^{-4}}{3a^{-2}b}$

Answer

A negative exponent is resolved by taking the reciprocal. For example $a^-2=\frac{1}{a^2}$ .

$\frac{6a^3b^{-4}}{3a^{-2}b}$

start by making all the negative exponents positive ones:

$\frac{6a^3}{3bb^4}\div \frac{1}{a^2}$ Note that the whole fraction on the left could have also been written as being divided by a^2 where the one is simply in the denominator, but it is necessary to understand that dividing by a fraction is the same as multiplying by one which occurs in the next step.

$\frac{6a^3a^5}{3b*b^4}$

Use the multiplication rule of exponents and simplify the constant:

$\frac{2a^5}{b^5}$

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Question

Simplify: $\frac{12j^{-4}k^2l^{-6}}{90j^8k^{-13}l^2}$

Answer

First, make all of the negative exponents positive. To do this, put it in the opposite location (if in the numerator, place in the denominator). This should look like: $\frac{12k^{13}k^2}{90j^4j^8l^2l^6}$ . Then, simplify each term. Remember, when multiplying and bases are the same, add exponents. Therefore, your final answer should be: $\frac{2k^{15}}{15j^{12}l^8}$ .

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Question

Evaluate $2^{-1}$

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

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

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Question

Evaluate $3^{-2}$

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{3^{2}}=\frac{1}{9}$

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Question

Evaluate $-4^{-3}$

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{-4^{3}}=-\frac{1}{64}$ Because the exponent is odd, that's why our fraction is negative.

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Question

Evaluate $-8^{-2}$

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{-8^2}=-\frac{1}{64}$ The reason the answer is negative is because we focus on the exponent first and in this case the exponent is raised to a positive .

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Question

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

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

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

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Question

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

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{\frac{1}{3}^{2}}=\frac{1}{\frac{1}{9}}=9$

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Question

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

Answer

When dealing with exponents, always turn it into this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{(-8)^{2}}=\frac{1}{64}$ It is important to keep the paranthesis as we are squaring which makes our answer.

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