Simplify `root3((1)/(8)xx(125)/(64))`

Options

A. `(5)/(8)`
B. `(375)/(512)`
C. `2(1)/(2)`
D. `15(5)/(8)`

Answer

To simplify the expression \( \sqrt[3]{\frac{1}{8} \times \frac{125}{64}} \), we can follow these steps:

### Step 1: Write the expression clearly
We start with the expression:
\[ \sqrt[3]{\frac{1}{8} \times \frac{125}{64}} \]

### Step 2: Combine the fractions
We can combine the fractions inside the cube root:
\[ \sqrt[3]{\frac{1 \times 125}{8 \times 64}} = \sqrt[3]{\frac{125}{512}} \]

### Step 3: Factor the numbers
Next, we can factor both the numerator and the denominator:
- \( 125 = 5^3 \)
- \( 512 = 8^3 = (2^3)^3 = 2^9 \)

So we can rewrite the expression as:
\[ \sqrt[3]{\frac{5^3}{2^9}} \]

### Step 4: Apply the cube root
Now we can apply the cube root to both the numerator and the denominator:
\[ \frac{\sqrt[3]{5^3}}{\sqrt[3]{2^9}} = \frac{5}{2^{9/3}} = \frac{5}{2^3} \]

### Step 5: Simplify the denominator
Calculating \( 2^3 \):
\[ 2^3 = 8 \]
So we have:
\[ \frac{5}{8} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{5}{8} \]