To determine which expression is equivalent to \( 5^{8.13} \), let's break down the exponent \( 8.13 \) into parts that can be more easily managed.
### Step-by-Step Solution:
1. Break Down the Exponent:
- The exponent \( 8.13 \) can be expressed as a sum of smaller parts:
[tex]\[
8.13 = 8 + 0.10 + 0.03
\][/tex]
2. Rewrite the Original Expression:
- Using the properties of exponents, specifically \( a^{b+c+d} = a^b \cdot a^c \cdot a^d \), we can rewrite the original expression \( 5^{8.13} \) as:
[tex]\[
5^{8.13} = 5^{8 + 0.10 + 0.03} = 5^8 \cdot 5^{0.10} \cdot 5^{0.03}
\][/tex]
3. Convert Decimal Exponents to Fractions:
- The decimals \( 0.10 \) and \( 0.03 \) can be converted to fractions:
[tex]\[
0.10 = \frac{1}{10} \quad \text{and} \quad 0.03 = \frac{3}{100}
\][/tex]
4. Substitute the Fractional Exponents Back In:
- Replacing \( 0.10 \) and \( 0.03 \) in the expression, we get:
[tex]\[
5^8 \cdot 5^{0.10} \cdot 5^{0.03} = 5^8 \cdot 5^{\frac{1}{10}} \cdot 5^{\frac{3}{100}}
\][/tex]
5. Identify the Correct Equivalent Expression:
- Out of the given options, we need to match this form:
- Option A: \( 5^8 \cdot 5^{13} \) – This is incorrect because it does not represent the breakdown of \( 8.13 \).
- Option C: \( 5^8 \cdot 5^{\frac{1}{10}} \cdot 5^{\frac{3}{100}} \) – This correctly represents the expression \( 5^{8.13} \).
Therefore, the correct equivalent expression is Option C.
So, the final answer is:
[tex]\(\boxed{C}\)[/tex]
