The value of `root(3)((7)/(875))` is equal to

Options

A. `(1)/(3)`
B. `(1)/(15)`
C. `(1)/(4)`
D. `(1)/(5)`

Answer

To find the value of \( \sqrt[3]{\frac{7}{875}} \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \sqrt[3]{\frac{7}{875}} = \left(\frac{7}{875}\right)^{\frac{1}{3}} \]

### Step 2: Simplify the Fraction
Next, we simplify the fraction \( \frac{7}{875} \). We can divide both the numerator and the denominator by 7:
\[ \frac{7}{875} = \frac{7 \div 7}{875 \div 7} = \frac{1}{125} \]

### Step 3: Substitute Back into the Expression
Now we substitute back into our expression:
\[ \left(\frac{1}{125}\right)^{\frac{1}{3}} \]

### Step 4: Rewrite 125 as a Power
We know that \( 125 = 5^3 \). Therefore, we can rewrite our expression as:
\[ \left(\frac{1}{5^3}\right)^{\frac{1}{3}} = \frac{1^{\frac{1}{3}}}{(5^3)^{\frac{1}{3}}} = \frac{1}{5^{3 \cdot \frac{1}{3}}} = \frac{1}{5^1} \]

### Step 5: Final Simplification
This simplifies to:
\[ \frac{1}{5} \]

Thus, the value of \( \sqrt[3]{\frac{7}{875}} \) is:
\[ \frac{1}{5} \]

### Final Answer
The final answer is \( \frac{1}{5} \).

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