To determine which option is equivalent to the expression [tex]\(5^{8.13}\)[/tex], let's break down the exponent [tex]\(8.13\)[/tex] into its component parts and utilize properties of exponents.
1. Break down the exponent:
The exponent [tex]\(8.13\)[/tex] can be seen as:
[tex]\[8.13 = 8 + 0.1 + 0.03\][/tex]
2. Rewrite the expression using the properties of exponents:
Using the property that [tex]\(a^{b+c} = a^b \cdot a^c\)[/tex], we can rewrite [tex]\(5^{8.13}\)[/tex] as:
[tex]\[5^{8.13} = 5^{8 + 0.1 + 0.03} = 5^8 \cdot 5^{0.1} \cdot 5^{0.03}\][/tex]
3. Match this with the given options:
Let's examine each option to see which matches the above expression:
- Option A: [tex]\(5^{8 \cdot \frac{1}{10} + \frac{3}{100}}\)[/tex]
This does not match our expression since it incorrectly combines the components in a different format.
- Option B: [tex]\(5^8 \cdot 5^{1 / 10} \cdot 5^{3 / 100}\)[/tex]
This matches our expression exactly, since [tex]\(5^{0.1}\)[/tex] is equivalent to [tex]\(5^{1 / 10}\)[/tex] and [tex]\(5^{0.03}\)[/tex] is equivalent to [tex]\(5^{3 / 100}\)[/tex].
- Option C: [tex]\(5^8 \cdot 5^{13 n 0}\)[/tex]
This expression does not match our breakdown and seems nonsensical in this context.
- Option D: [tex]\(5^8 + 5^{1 / 10} + 5^{3100}\)[/tex]
This expression uses addition instead of multiplication, which is incorrect for exponents.
Therefore, the correct answer is:
B. [tex]\(5^8 \cdot 5^{1 / 10} \cdot 5^{3 / 100}\)[/tex]
