The value of `25^((-1//4log_(5)25))` is equal to

Options

A. `1/5`
B. `-1/25`
C. `-25`
D. None of these

Answer

To solve the expression \( 25^{\left(-\frac{1}{4} \log_{5} 25\right)} \), we will follow these steps:

### Step 1: Rewrite the base
We know that \( 25 \) can be expressed as \( 5^2 \). Therefore, we can rewrite the expression as:
\[ (5^2)^{\left(-\frac{1}{4} \log_{5} 25\right)} \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can simplify the expression:
\[ 5^{2 \cdot \left(-\frac{1}{4} \log_{5} 25\right)} = 5^{-\frac{1}{2} \log_{5} 25} \]

### Step 3: Simplify the logarithm
Next, we need to simplify \(\log_{5} 25\). Since \(25 = 5^2\), we can use the property of logarithms:
\[ \log_{5} 25 = \log_{5} (5^2) = 2 \]

### Step 4: Substitute back into the expression
Now we substitute \(\log_{5} 25\) back into our expression:
\[ 5^{-\frac{1}{2} \cdot 2} = 5^{-1} \]

### Step 5: Evaluate the final expression
Finally, we can evaluate \(5^{-1}\):
\[ 5^{-1} = \frac{1}{5} \]

Thus, the value of \( 25^{\left(-\frac{1}{4} \log_{5} 25\right)} \) is:
\[ \frac{1}{5} \]

### Final Answer:
\[ \frac{1}{5} \]