$$ \left. \begin{array} { l } { 64 ^ { - \frac { 2 } { 3 } } = \square } \ { ( \frac { 1 } { 125 } ) ^ { \frac { 2 } { 3 } } = \square } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 64 ^ { - \frac { 2 } { 3 } } = \square } \ { ( \frac { 1 } { 125 } ) ^ { \frac { 2 } { 3 } } = \square } \end{array} ight. $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{1}{125}\right)^{\frac{2}{3}}$$
- step1: Use the properties of exponents:
  $$\frac{1^{\frac{2}{3}}}{125^{\frac{2}{3}}}$$
- step2: Evaluate the power:
  $$\frac{1}{125^{\frac{2}{3}}}$$
- step3: Evaluate the power:
  $$\frac{1}{25}$$
Calculate or simplify the expression $$ 64 ^ { - \frac { 2 } { 3 } } $$.
Calculate the value by following steps:
- step0: Calculate:
  $$64^{-\frac{2}{3}}$$
- step1: Express with a positive exponent:
  $$\frac{1}{64^{\frac{2}{3}}}$$
- step2: Evaluate the power:
  $$\frac{1}{16}$$
Let's solve the two expressions step by step.

1. **For the first expression**: 
   $$
   64^{-\frac{2}{3}}
   $$
   - First, we can rewrite $$64$$ as $$4^3$$ since $$4^3 = 64$$.
   - Thus, we have:
   $$
   (4^3)^{-\frac{2}{3}} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}
   $$
   - In decimal form, this is $$0.0625$$.

2. **For the second expression**:
   $$
   \left(\frac{1}{125}\right)^{\frac{2}{3}}
   $$
   - We can rewrite $$125$$ as $$5^3$$ since $$5^3 = 125$$.
   - Therefore, we have:
   $$
   \left(\frac{1}{5^3}\right)^{\frac{2}{3}} = \frac{1^{\frac{2}{3}}}{(5^3)^{\frac{2}{3}}} = \frac{1}{5^2} = \frac{1}{25}
   $$
   - In decimal form, this is $$0.04$$.

So, the final results are:
- $$64^{-\frac{2}{3}} = \frac{1}{16} \text{ or } 0.0625$$
- $$\left(\frac{1}{125}\right)^{\frac{2}{3}} = \frac{1}{25} \text{ or } 0.04$$
$$64^{-\frac{2}{3}} = \frac{1}{16}$$ and $$\left(\frac{1}{125}\right)^{\frac{2}{3}} = \frac{1}{25}$$.

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