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Question 16 of 41
Which of the following is equivalent to the expression below?
[tex]\[ 5^{8.13} \][/tex]

A. \( 5^8 \cdot 5^{0.13} \)

B. \( 5^3 \cdot 5^{1/10} \cdot 5^{3/100} \)

C. \( 5^{8+0.13} \)

D. [tex]\( 5^{8} \cdot 5^{0.1+0.03} \)[/tex]


Sagot :

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]