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((64)/(b^(27)))^(-(2)/(3)) Which of the following is equivalent to the given expression? Choose 1 answer: (A) (b^(9))/(16) (B) (b^(18))/(16) (C) -(128b^(9))/(3) (D) -(128b^(18))/(3)

(64b27)−23 \left(\frac{64}{b^{27}}\right)^{-\frac{2}{3}} \newlineWhich of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) b916 \frac{b^{9}}{16} \newline(B) b1816 \frac{b^{18}}{16} \newline(C) −128b93 -\frac{128 b^{9}}{3} \newline(D) −128b183 -\frac{128 b^{18}}{3}

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Q. (64b27)−23 \left(\frac{64}{b^{27}}\right)^{-\frac{2}{3}} \newlineWhich of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) b916 \frac{b^{9}}{16} \newline(B) b1816 \frac{b^{18}}{16} \newline(C) −128b93 -\frac{128 b^{9}}{3} \newline(D) −128b183 -\frac{128 b^{18}}{3}
  1. Simplify expression using exponents: Simplify the given expression using the properties of exponents.\newlineThe expression (64b27)−23(\frac{64}{b^{27}})^{-\frac{2}{3}} can be rewritten by applying the negative exponent rule, which states that a−n=1ana^{-n} = \frac{1}{a^n}. This gives us:\newline1(64b27)23\frac{1}{(\frac{64}{b^{27}})^{\frac{2}{3}}}
  2. Apply negative exponent rule: Simplify the expression inside the parentheses.\newlineThe expression inside the parentheses is a fraction raised to a power. We can apply the exponent to both the numerator and the denominator separately:\newline(1/((642/3)/(b27⋅(2/3))))(1/((64^{2/3})/(b^{27\cdot(2/3)})))
  3. Simplify expression inside parentheses: Calculate the exponent for the numerator.\newline642364^{\frac{2}{3}} is the cube root of 6464 squared. Since 6464 is 434^3, we have:\newline(43)23=43⋅(23)=42=16(4^3)^{\frac{2}{3}} = 4^{3\cdot(\frac{2}{3})} = 4^2 = 16
  4. Calculate numerator exponent: Calculate the exponent for the denominator.\newlineb27(23)b^{27\left(\frac{2}{3}\right)} is bb raised to the power of 2727 times 23\frac{2}{3}, which simplifies to:\newlineb18b^{18}
  5. Calculate denominator exponent: Combine the simplified numerator and denominator.\newlineNow we have:\newline1(16b18)=b1816\frac{1}{\left(\frac{16}{b^{18}}\right)} = \frac{b^{18}}{16}
  6. Combine numerator and denominator: Match the simplified expression with the given choices.\newlineThe expression b18/16b^{18}/16 corresponds to choice (B).

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