Let's simplify the expression [tex]\(5^3 \times 5^{-5}\)[/tex].
1. Apply the properties of exponents: One of the properties of exponents is that [tex]\(a^m \times a^n = a^{m+n}\)[/tex]. In this case, we apply this property to combine the exponents:
[tex]\[
5^3 \times 5^{-5} = 5^{3 + (-5)}
\][/tex]
2. Simplify the exponent: Add the exponents [tex]\(3\)[/tex] and [tex]\(-5\)[/tex]:
[tex]\[
3 + (-5) = -2
\][/tex]
3. Expression with a single exponent: Now the expression simplifies to:
[tex]\[
5^{-2}
\][/tex]
4. Interpret the negative exponent: Recall that a negative exponent signifies the reciprocal of the base raised to the absolute value of the exponent:
[tex]\[
5^{-2} = \frac{1}{5^2}
\][/tex]
5. Calculate the value of [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
6. Final expression: So, the expression [tex]\(5^{-2}\)[/tex] can be written as:
[tex]\[
5^{-2} = \frac{1}{25}
\][/tex]
Therefore, the simplified form of the expression [tex]\(5^3 \times 5^{-5}\)[/tex] is [tex]\(\frac{1}{5^2}\)[/tex].
So, the correct answer is:
B. [tex]\(\frac{1}{5^2}\)[/tex]
