Solving a Negative Fractional Exponent Expression
The expression in the image is \((\frac{8}{27})^{-2/3}\).
To solve this, we can apply the rule for negative exponents and fractional exponents. A negative exponent means that you take the reciprocal of the base, and a fractional exponent means you take the root of the base (the denominator of the fraction) and then raise it to the power of the numerator.
Here's the step-by-step calculation:
\[
\left(\frac{8}{27}\right)^{-2/3} = \left(\frac{27}{8}\right)^{2/3}
\]
Now we take the cube root of both the numerator and the denominator and then square them because the exponent is \(2/3\):
\[
\left(\sqrt[3]{\frac{27}{8}}\right)^2 = \left(\frac{\sqrt[3]{27}}{\sqrt[3]{8}}\right)^2 = \left(\frac{3}{2}\right)^2
\]
Finally, we square the result:
\[
\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}
\]
Therefore, \((\frac{8}{27})^{-2/3}\) equals \(\frac{9}{4}\) or 2.25.
