Find the value of (27/8)^-2/3 give your answer as a fraction in its simplest form
The Correct Answer and Explanation is :
To solve (\left(\frac{27}{8}\right)^{-\frac{2}{3}}), we will break it down step-by-step.
Step 1: Understanding the negative exponent
The negative exponent indicates that we should take the reciprocal of the base. So:
[
\left(\frac{27}{8}\right)^{-\frac{2}{3}} = \frac{1}{\left(\frac{27}{8}\right)^{\frac{2}{3}}}
]
Now, we need to evaluate (\left(\frac{27}{8}\right)^{\frac{2}{3}}).
Step 2: Simplifying the fractional exponent
The fractional exponent (\frac{2}{3}) can be interpreted as follows:
[
\left(\frac{27}{8}\right)^{\frac{2}{3}} = \left(\left(\frac{27}{8}\right)^{\frac{1}{3}}\right)^2
]
This means we first take the cube root of (\frac{27}{8}) and then square the result.
Step 3: Taking the cube root of (\frac{27}{8})
To find the cube root of a fraction, take the cube root of the numerator and the denominator separately:
- The cube root of 27 is 3 ((3^3 = 27)).
- The cube root of 8 is 2 ((2^3 = 8)).
So:
[
\left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{3}{2}
]
Step 4: Squaring the result
Now, we square (\frac{3}{2}):
[
\left(\frac{3}{2}\right)^2 = \frac{9}{4}
]
Thus:
[
\left(\frac{27}{8}\right)^{\frac{2}{3}} = \frac{9}{4}
]
Step 5: Taking the reciprocal
Going back to our original expression, we now take the reciprocal of (\frac{9}{4}):
[
\frac{1}{\frac{9}{4}} = \frac{4}{9}
]
Final Answer:
[
\left(\frac{27}{8}\right)^{-\frac{2}{3}} = \frac{4}{9}
]
Explanation Summary:
We used the negative exponent rule to take the reciprocal of the base. Then, we simplified the fractional exponent by first taking the cube root of (\frac{27}{8}), which gave (\frac{3}{2}), and then squaring this result. Finally, we found the reciprocal of (\frac{9}{4}) to get the final answer of (\frac{4}{9}).