Question

Simplify each expression. $$\left(-\frac{64}{125}\right)^{-2 / 3}$$

Short answer

\(\frac{16}{25}\)

Step-by-step solution

- Rewrite the Expression with Positive Exponent

Recall that a negative exponent means the reciprocal of the base with a positive exponent: \ \ \ \ \ \ \(\begin{aligned} \ \left(- \frac{64}{125}\right)^{-2 / 3} = \left(- \frac{64}{125}\right)^{2 / 3} \ \ \end{aligned}\)

- Apply the Rational Exponent Rule

Rewrite the expression using the property of rational exponents: \ \ \ \ \ \ \(\begin{aligned} \ \left(\frac{-64}{125}\right)^{2 / 3} = \ \left(\left(\frac{-64}{125}\right)^{1 / 3}\right)^2 \ \ \end{aligned}\)

- Simplify the Cube Root

Evaluate the cube root of the fraction: \ \ \ \ \ \ \(\begin{aligned} \ \left(\frac{-64}{125}\right)^{1 / 3} = \frac{\sqrt[3]{-64}}{\sqrt[3]{125} \ \ \ \ \ \ \sqrt[3]{-64}= \ -4\ \ \sqrt[3]{125}= \5 \ \ \ \ \ \ \ \left(\frac{-64}{125}\right)^{1 / 3}= \ \frac{-4}{5} \end{aligned}\)

- Square the Result of the Cube Root

Raise the simplified cube root to the power of 2: \ \ \ \ \ \ \(\begin{aligned} \ \left(\frac{-4}{5}\right)^2 = \frac{16}{25} \d \end{aligned}\)

Key concepts

Negative Exponents

Negative exponents might seem tricky, but they simply represent the reciprocal of the base number raised to the corresponding positive exponent. For example, if you see \(\begin{aligned} a^{-b} \end{aligned}\), it means \(\begin{aligned} \frac{1}{a^b} \end{aligned}\).
In our problem, we start with the expression \(\begin{aligned} \left(- \frac{64}{125} \right)^{-2 / 3} \end{aligned}\).
To convert the negative exponent to a positive one, we take the reciprocal of \(\begin{aligned} - \frac{64}{125} \end{aligned}\) and then apply the positive exponent. This results in: \(\begin{aligned} \left( - \frac{64}{125} \right)^{-2/3} = \left( - \frac{125}{64} \right)^{2/3} \end{aligned}\).
By flipping the fraction and removing the negative exponent, we can proceed to apply further simplification steps.

Rational Exponents

Rational exponents are another way to express roots. A rational exponent like \(\begin{aligned} \frac{m}{n} \end{aligned}\) means we are taking the nth root of a number and then raising it to the mth power.
For instance, \(\begin{aligned} a^{\frac{m}{n}} \end{aligned}\) equals \(\begin{aligned} \left( \sqrt[n]{a} \right)^m \end{aligned}\).
Using this knowledge in our problem, we rewrite \(\begin{aligned} \left( - \frac{64}{125} \right)^{2/3} \end{aligned}\) as \(\begin{aligned} \left( \begin{bmatrix} \left( \frac{-64}{125} \right)^{1/3} \end{bmatrix} \right)^2 \end{aligned}\).
This step helps to visualize the expression better by separating the root and the power.

Cube Roots

Cube roots are simply finding a number which, when multiplied by itself three times, gives the original number.
The cube root is denoted as \(\begin{aligned} \sqrt[3]{x} \end{aligned}\).
For instance, \(\begin{aligned} \sqrt[3]{8} = 2 \end{aligned}\) because \(\begin{aligned} 2 \times 2 \times2 = 8 \end{aligned}\).
Let's apply this to our problem. We need to evaluate the cube roots for \(\begin{aligned} \sqrt[3]{-64} \end{aligned}\) and \(\begin{aligned} \sqrt[3]{125} \end{aligned}\).
We find \(\begin{aligned} \sqrt[3]{-64}= -4 \end{aligned}\) because \(\begin{aligned} -4 \times -4 \times -4 = -64 \end{aligned}\),
and \(\begin{aligned} \sqrt[3]{125}= 5 \end{aligned}\) because \(\begin{aligned} 5 \times 5 \times 5 = 125 \end{aligned}\).
Putting these results back in the expression gives us \(\begin{aligned} \frac{-4}{5} \end{aligned}\).
The last step involves squaring the fraction, resulting in \(\begin{aligned} \left( \frac{-4}{5} \right)^2 = \frac{16}{25} \end{aligned}\).