Question

25³·5⁵ / 5²

Explanation:

Step1: Rewrite 25 as 5²

Since \( 25 = 5^2 \), we can rewrite the numerator's first term: \( 25^3=(5^2)^3 \).
Using the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we get \((5^2)^3 = 5^{2\times3}=5^6\).
So the numerator becomes \(5^6\times5^5\).

Step2: Use the product rule for exponents

The product rule for exponents states that \(a^m\times a^n=a^{m + n}\). For \(a = 5\), \(m = 6\) and \(n = 5\), we have \(5^6\times5^5=5^{6 + 5}=5^{11}\).
Now our fraction is \(\frac{5^{11}}{5^2}\).

Step3: Use the quotient rule for exponents

The quotient rule for exponents is \(a^m\div a^n=a^{m - n}\) (where \(a
eq0\)). For \(a = 5\), \(m = 11\) and \(n = 2\), we get \(5^{11-2}=5^9\).

Answer:

\(5^9\) (or 1953125, but in exponential form \(5^9\) is more appropriate for this problem)