Express `5^(8)` in exponent form with base 25.

Answer

To express \( 5^8 \) in exponent form with base 25, we can follow these steps:

### Step 1: Rewrite the base 25 in terms of base 5
We know that \( 25 \) can be expressed as \( 5^2 \). Therefore, we can write:
\[ 25 = 5^2 \]

### Step 2: Substitute the base in the expression
Now, we can substitute \( 25 \) in the expression \( 5^8 \) as follows:
\[ 5^8 = (5^2)^n \]
where \( n \) is the exponent we need to find.

### Step 3: Use the property of exponents
According to the property of exponents, \( (a^m)^n = a^{m \cdot n} \). Thus, we can rewrite our expression:
\[ (5^2)^n = 5^{2n} \]

### Step 4: Set the exponents equal to each other
Since both expressions represent the same quantity, we can set the exponents equal to each other:
\[ 2n = 8 \]

### Step 5: Solve for \( n \)
Now, we can solve for \( n \):
\[ n = \frac{8}{2} = 4 \]

### Step 6: Write the final expression
Now that we have \( n \), we can express \( 5^8 \) in terms of base 25:
\[ 5^8 = (5^2)^4 = 25^4 \]

Thus, the final answer is:
\[ 5^8 = 25^4 \]

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