Questions: Evaluate. Write your answer as a whole number or [ frac5^2 cdot 5^-1 cdot 5^-65^-8= ]

Solution Steps

To solve the given expression, we need to use the rules of exponents. Specifically, we will use the following rules:

1. \(a^m \cdot a^n = a^{m+n}\)
2. \(\frac{a^m}{a^n} = a^{m-n}\)

First, combine the exponents in the numerator using the multiplication rule. Then, simplify the fraction by subtracting the exponent in the denominator from the combined exponent in the numerator.

Using the rule \(a^m \cdot a^n = a^{m+n}\), we combine the exponents in the numerator: \[ 5^2 \cdot 5^{-1} \cdot 5^{-6} = 5^{2 + (-1) + (-6)} = 5^{-5} \]

Using the rule \(\frac{a^m}{a^n} = a^{m-n}\), we simplify the fraction: \[ \frac{5^{-5}}{5^{-8}} = 5^{-5 - (-8)} = 5^{-5 + 8} = 5^3 \]

Calculate \(5^3\): \[ 5^3 = 125 \]

Final Answer