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]
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]
