if `x=root3(2(93)/125)`, then the value of x is

Options

A. `2(1)/5`
B. `1(2)/5`
C. `3(4)/5`
D. `4(1)/5`

Answer

To solve the problem where \( x = \sqrt[3]{\frac{2^{93}}{125}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ x = \sqrt[3]{\frac{2^{93}}{125}} \]

2. **Recognize that 125 is a power of 5**:
\[ 125 = 5^3 \]
So we can rewrite the expression as:
\[ x = \sqrt[3]{\frac{2^{93}}{5^3}} \]

3. **Split the cube root**:
Using the property of cube roots, we can separate the numerator and denominator:
\[ x = \frac{\sqrt[3]{2^{93}}}{\sqrt[3]{5^3}} \]

4. **Simplify the denominator**:
The cube root of \( 5^3 \) is simply 5:
\[ x = \frac{\sqrt[3]{2^{93}}}{5} \]

5. **Simplify the numerator**:
The cube root of \( 2^{93} \) can be simplified using the property of exponents:
\[ \sqrt[3]{2^{93}} = 2^{93/3} = 2^{31} \]
Thus, we have:
\[ x = \frac{2^{31}}{5} \]

6. **Final value of x**:
Therefore, the value of \( x \) is:
\[ x = \frac{2^{31}}{5} \]