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

Question

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

Bytelearn · Accepted Answer

Understanding exponent properties: Understand the properties of exponents that will be used to solve the problem. The properties of exponents that are relevant here are: (a^m)^n = a^(m*n) - Raising a power to a power means you multiply the exponents. a^(-n) = 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))^((1)/(9)). Using the property (a^m)^n = a^(m*n), we get: (v^(-1))^((1)/(9)) = v^((-1)*(1/9)) = v^(-(1)/(9))Comparing the result: Compare the result from Step 2 with the given choices. We found that (v^(-1))^((1)/(9)) simplifies to v^(-(1)/(9)). Now we need to check which of the given choices are equivalent to this expression.Evaluating choice A: Evaluate choice A: (root(9)(v))^(-1). The expression (root(9)(v))^(-1) is equivalent to (v^(1/9))^(-1) because the 9th 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. Since we have already found that choices A, B, and C are equivalent to the original expression, choice D is not correct.

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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