Select the correct answer.

Simplify the expression [tex]5^3 \times 5^{-5}[/tex]

A. [tex]\frac{1}{5^2}[/tex]

B. [tex]\frac{1}{5}[/tex]

C. [tex]-5^2[/tex]

D. [tex]5^2[/tex]



Answer :

To simplify the expression \(5^3 \times 5^{-5}\), let's go through the steps methodically:

1. Apply the property of exponents: When multiplying numbers with the same base, we can add the exponents. The property is given by:
[tex]\[ a^m \times a^n = a^{m+n} \][/tex]

2. Combine the exponents: Using this property, we can combine the exponents of 5 in the expression \(5^3 \times 5^{-5}\):
[tex]\[ 5^3 \times 5^{-5} = 5^{3 + (-5)} = 5^{-2} \][/tex]

3. Interpret the negative exponent: A negative exponent indicates that the base is on the wrong side of a fraction line. Specifically, \(a^{-n} = \frac{1}{a^n}\). Therefore,
[tex]\[ 5^{-2} = \frac{1}{5^2} \][/tex]

Through these steps, we find that \(5^3 \times 5^{-5}\) simplifies to \(\frac{1}{5^2}\).

Thus, the correct answer is:
A. [tex]\(\frac{1}{5^2}\)[/tex]