The radical form of `((13)/(25))^(3//4)` is

Options

A. `root3(((13)/(25))^(4))`
B. `root4(((13)/(25))^(3))`
C. `root4(((25)/(13))^(3))`
D. `root3(((25)/(13))^(4))`

Answer

To convert the expression \(\left(\frac{13}{25}\right)^{\frac{3}{4}}\) into its radical form, we will follow these steps:

### Step 1: Understand the expression
The expression \(\left(\frac{13}{25}\right)^{\frac{3}{4}}\) can be interpreted as raising the fraction \(\frac{13}{25}\) to the power of \(\frac{3}{4}\).

### Step 2: Rewrite the exponent
The exponent \(\frac{3}{4}\) can be separated into two parts: the numerator (3) and the denominator (4). This means we can express the exponent as follows:
\[ \left(\frac{13}{25}\right)^{\frac{3}{4}} = \left(\frac{13}{25}\right)^{3} \cdot \left(\frac{13}{25}\right)^{\frac{1}{4}} \]

### Step 3: Apply the radical form
The term \(\left(\frac{13}{25}\right)^{\frac{1}{4}}\) can be expressed in radical form as the fourth root:
\[ \left(\frac{13}{25}\right)^{\frac{1}{4}} = \sqrt[4]{\frac{13}{25}} \]
Thus, we can rewrite the original expression as:
\[ \left(\frac{13}{25}\right)^{\frac{3}{4}} = \left(\frac{13}{25}\right)^{3} \cdot \sqrt[4]{\frac{13}{25}} \]

### Step 4: Calculate \(\left(\frac{13}{25}\right)^{3}\)
Now, we need to calculate \(\left(\frac{13}{25}\right)^{3}\):
\[ \left(\frac{13}{25}\right)^{3} = \frac{13^3}{25^3} = \frac{2197}{15625} \]

### Step 5: Combine the results
Now, we can combine the results:
\[ \left(\frac{13}{25}\right)^{\frac{3}{4}} = \frac{2197}{15625} \cdot \sqrt[4]{\frac{13}{25}} \]

### Final Result
Thus, the radical form of \(\left(\frac{13}{25}\right)^{\frac{3}{4}}\) is:
\[ \frac{2197}{15625} \cdot \sqrt[4]{\frac{13}{25}} \]