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Which expressions are equivalent to
(v^(-1))^((1)/(9)) ?
Choose all answers that apply:
A
(root(9)(v))^(-1)
B
(v^((1)/(9)))^(-1)
C
v^(-(1)/(9))
D None of the above

Which expressions are equivalent to $\left(v^{-1}\right)^{\frac{1}{9}}$ ? $\newline$ Choose all answers that apply: $\newline$ A $(\sqrt[9]{v})^{-1}$ $\newline$ B $\left(v^{\frac{1}{9}}\right)^{-1}$ $\newline$ C $v^{-\frac{1}{9}}$ $\newline$ D None of the above

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Compare linear and exponential growth

Full solution

Q. Which expressions are equivalent to

\left(v^{-1}\right)^{\frac{1}{9}}

?

\newline

Choose all answers that apply:

\newline

A

(\sqrt[9]{v})^{-1}

\newline

B

\left(v^{\frac{1}{9}}\right)^{-1}

\newline

C

v^{-\frac{1}{9}}

\newline

D None of the above

Understanding exponent properties: Understand the properties of exponents that will be used to solve the problem. $\newline$ The properties of exponents that are relevant here are: $\newline$ $(a^m)^n = a^{(m*n)}$ - Raising a power to a power means you multiply the exponents. $\newline$ $a^{-n} = \frac{1}{a^n}$ - A negative exponent means that the base is on the opposite side of the fraction line.
Applying exponent property to given expression: Apply the exponent property to the given expression $(v^{-1})^{\frac{1}{9}}$ . Using the property $(a^m)^n = a^{m*n}$ , we get: $(v^{-1})^{\frac{1}{9}} = v^{(-1)*(\frac{1}{9})} = v^{-\frac{1}{9}}$
Comparing the result: Compare the result from Step $2$ with the given choices. $\newline$ We found that $(v^{-1})^{\frac{1}{9}}$ simplifies to $v^{-\frac{1}{9}}$ . $\newline$ Now we need to check which of the given choices are equivalent to this expression.
Evaluating choice A: Evaluate choice A: $(\sqrt[9]{v})^{-1}$ . The expression $(\sqrt[9]{v})^{-1}$ is equivalent to $(v^{1/9})^{-1}$ because the $9$ th root of $v$ is $v$ raised to the $1/9$ power. Using the property $(a^m)^n = a^{m*n}$ , we get: $(v^{1/9})^{-1} = v^{(1/9)*(-1)} = v^{-(1)/(9)}$ This matches the expression we found in Step $2$ .
Evaluating choice B: Evaluate choice B: $(v^{(1/9)})^{-1}$ . This is the same expression we evaluated in Step $4$ , and we have already determined that it is equivalent to $v^{-(1/9)}$ .
Evaluating choice C: Evaluate choice C: $v^{-(1)/(9)}$ . This is exactly the expression we derived in Step $2$ , so it is clearly equivalent to $(v^{-1})^{(1)/(9)}$ .
Evaluating choice D: Evaluate choice D: None of the above. $\newline$ Since we have already found that choices A, B, and C are equivalent to the original expression, choice D is not correct.

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\newline

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\newline

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\newline

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