To simplify the expression [tex]\( 5^3 \times 5^7 \)[/tex], we use the laws of exponents.
1. Law of Exponents: When you multiply two expressions that have the same base, you add their exponents. Mathematically, this is written as:
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
where [tex]\( a \)[/tex] is the base and [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the exponents.
2. Identify the components of the problem:
- Base: [tex]\( 5 \)[/tex]
- Exponents: [tex]\( 3 \)[/tex] and [tex]\( 7 \)[/tex]
3. Apply the law of exponents:
[tex]\[
5^3 \times 5^7 = 5^{3+7}
\][/tex]
4. Add the exponents:
[tex]\[
3 + 7 = 10
\][/tex]
5. The simplified expression is:
[tex]\[
5^{10}
\][/tex]
Next, we see if this matches any of the given multiple-choice options:
A. [tex]\(\frac{1}{5^2}\)[/tex] [tex]$\quad$[/tex] (incorrect, because it simplifies to [tex]\( 5^{-2} \)[/tex])
B. [tex]\(5^2\)[/tex] [tex]$\quad$[/tex] (incorrect, because [tex]\( 5^{10} \neq 5^2 \)[/tex])
C. [tex]\(\frac{1}{5}\)[/tex] [tex]$\quad$[/tex] (incorrect, because it simplifies to [tex]\( 5^{-1} \)[/tex])
D. [tex]\(-5^2\)[/tex] [tex]$\quad$[/tex] (incorrect, because [tex]\( 5^{10} \neq -5^2 \)[/tex])
The simplified expression [tex]\( 5^{10} \)[/tex] does not match any of the given multiple-choice options directly. But considering the phrasing of the question specifically aimed at simplification, we've confirmed our final result:
The simplified expression for [tex]\( 5^3 \times 5^7 \)[/tex] is [tex]\( 5^{10} \)[/tex].
Hence, the answer is not listed directly in the choices provided, but the simplification process is correct.
