The value of `""_(25)^(-1//4 log_(5)(25))` is:

Options

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

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ 25^{-\frac{1}{4} \log_{5}(25)} \]

### Step 2: Change the base of the logarithm
We know that \( 25 \) can be expressed as \( 5^2 \). Therefore, we can rewrite \( \log_{5}(25) \) as:
\[ \log_{5}(25) = \log_{5}(5^2) = 2 \]

### Step 3: Substitute the value of the logarithm
Now, we substitute \( \log_{5}(25) \) back into the expression:
\[ 25^{-\frac{1}{4} \cdot 2} \]

### Step 4: Simplify the exponent
Calculating the exponent:
\[ -\frac{1}{4} \cdot 2 = -\frac{2}{4} = -\frac{1}{2} \]
Thus, our expression simplifies to:
\[ 25^{-\frac{1}{2}} \]

### Step 5: Rewrite \( 25 \) in terms of base \( 5 \)
Again, we know \( 25 = 5^2 \), so we can rewrite the expression:
\[ (5^2)^{-\frac{1}{2}} \]

### Step 6: Apply the power of a power property
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):
\[ 5^{2 \cdot -\frac{1}{2}} = 5^{-1} \]

### Step 7: Simplify the expression
Finally, we can express \( 5^{-1} \) as:
\[ \frac{1}{5} \]

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

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