The value of `(0.125)^(2//3)` is

Options

A. `2.5 `
B. `0.25 `
C. `0.025 `
D. `0.0025 `

Answer

To find the value of \( (0.125)^{\frac{2}{3}} \), we can follow these steps:

### Step 1: Rewrite 0.125 as a Fraction
First, we rewrite \( 0.125 \) as a fraction.
\[ 0.125 = \frac{125}{1000} \]

### Step 2: Simplify the Fraction
Next, we simplify \( \frac{125}{1000} \).
\[ \frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \]

### Step 3: Rewrite the Expression
Now, we can rewrite the original expression using the simplified fraction:
\[ (0.125)^{\frac{2}{3}} = \left(\frac{1}{8}\right)^{\frac{2}{3}} \]

### Step 4: Apply the Power to the Fraction
Using the property of exponents, we can apply the power to both the numerator and the denominator:
\[ \left(\frac{1}{8}\right)^{\frac{2}{3}} = \frac{1^{\frac{2}{3}}}{8^{\frac{2}{3}}} \]

### Step 5: Simplify the Numerator
Since \( 1^{\frac{2}{3}} = 1 \):
\[ \frac{1}{8^{\frac{2}{3}}} \]

### Step 6: Calculate \( 8^{\frac{2}{3}} \)
To calculate \( 8^{\frac{2}{3}} \), we first find the cube root of \( 8 \) and then square it:
\[ 8 = 2^3 \implies 8^{\frac{1}{3}} = 2 \]
Now square it:
\[ 8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4 \]

### Step 7: Final Calculation
Now substituting back, we have:
\[ (0.125)^{\frac{2}{3}} = \frac{1}{4} \]

### Conclusion
Thus, the value of \( (0.125)^{\frac{2}{3}} \) is:
\[ \frac{1}{4} = 0.25 \]

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