Correct Answer
three fourths
$$ \large{\frac{3}{4}} $$
Explanation
To rewrite open parenthesis fraction with numerator of 64 and denominator of 27 close parenthesis raised to the negative one third power \( \left(\frac{64}{27}\right)^{-\frac{1}{3}} \) as a fraction, we can follow a series of steps that involve the laws of exponents and the properties of radicals. Let's break it down step by step.
First, we apply the negative exponent rule, which states that a to the negative n power equals fraction with numerator of 1 and denominator of a to the n power. \( a^{-n} = \frac{1}{a^n}. \) So, we can rewrite open parenthesis fraction with numerator of 64 and denominator of 27 close parenthesis, raised to the negative one-third power \( \left(\frac{64}{27}\right)^{-\frac{1}{3}} \) as:open parenthesis fraction with numerator of 64 and denominator of 27 close parenthesis raised to the negative one-third power equals fraction with numerator of 1 and denominator of open parenthesis fraction with numerator of 64 and denominator of 27 close parenthesis raised to the one-third power, end fraction.
$$ \left(\frac{64}{27}\right)^{-\frac{1}{3}} = \frac{1}{\left(\frac{64}{27}\right)^{\frac{1}{3}}} $$
The cube root (fractional exponent) rule states that the exponent one-third \( \frac{1}{3} \) means we need to take the cube root of fraction with numerator of 64 and denominator of 27. \( \frac{64}{27}. \) The cube root of a fraction is the same as taking the cube root of the numerator and denominator separately:fraction with numerator of 1 and denominator of open parenthesis fration with numerator of 64 and denominator of 27 close parenthesis raised to the one-third power, end fraction, equals fraction with numerator of 1 and denominator of fraction with numerator of the cube root of 64 and denominator of the cube root of 27, end fraction.
$$ \frac{1}{\left(\frac{64}{27}\right)^{\frac{1}{3}}} = \frac{1}{\frac{\sqrt[3]{64}}{\sqrt[3]{27}}} $$
The cube root of 64 is 4, and the cube root of 27 is 3, so we now have, after simplifying the reciprocal:fraction with numerator of 1 and denominator of fraction with numerator of the cube root of 64 and denominator of the cube root of 27, end fraction, equals fraction with numerator of 1 and denominator of four thirds equals three fourths.
$$ \frac{1}{\frac{\sqrt[3]{64}}{\sqrt[3]{27}}} = \frac{1}{\frac{4}{3}} = \frac{3}{4} $$
