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Which expressions are equivalent to (d^((1)/(8)))^(5) ? Choose all answers that apply: A. (d^(5))^((1)/(8)) B (d^(5))^(8) C (root(8)(d))^(5) D None of the above

Which expressions are equivalent to (d18)5 \left(d^{\frac{1}{8}}\right)^{5} ?\newlineChoose all answers that apply:\newlineA. (d5)18 \left(d^{5}\right)^{\frac{1}{8}} \newlineB (d5)8 \left(d^{5}\right)^{8} \newlineC (d8)5 (\sqrt[8]{d})^{5} \newlineD None of the above

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Q. Which expressions are equivalent to (d18)5 \left(d^{\frac{1}{8}}\right)^{5} ?\newlineChoose all answers that apply:\newlineA. (d5)18 \left(d^{5}\right)^{\frac{1}{8}} \newlineB (d5)8 \left(d^{5}\right)^{8} \newlineC (d8)5 (\sqrt[8]{d})^{5} \newlineD None of the above
  1. Understanding Exponent Properties: Understand the properties of exponents. When raising a power to a power, you multiply the exponents. This is based on the property (am)n=am∗n(a^m)^n = a^{m*n}.
  2. Applying Exponent Property to the Expression: Apply the exponent property to the given expression.\newline(d18)5=d(18⋅5)=d58(d^{\frac{1}{8}})^5 = d^{(\frac{1}{8}\cdot5)} = d^{\frac{5}{8}}
  3. Checking Answer Choice A: Check each answer choice to see if it is equivalent to d5/8d^{5/8}.
    A. (d5)1/8=d5∗(1/8)=d5/8(d^5)^{1/8} = d^{5*(1/8)} = d^{5/8}
    This matches our expression from Step 22, so choice A is equivalent.
  4. Checking Answer Choice B: Check choice B.\newlineB. (d5)8=d(5×8)=d40(d^5)^8 = d^{(5\times8)} = d^{40}\newlineThis does not match our expression from Step 22, so choice B is not equivalent.
  5. Checking Answer Choice C: Check choice C.\newlineC. (d8)5=(d1/8)5=d5/8(\sqrt[8]{d})^5 = (d^{1/8})^5 = d^{5/8}\newlineThis matches our expression from Step 22, so choice C is equivalent.
  6. Checking Answer Choice D: Check choice D.\newlineD. None of the above\newlineSince we have found that choices A and C are equivalent to the given expression, choice D is not correct.

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