Problem 126 - College Algebra

Question

Explain the quotient rule for exponents. Use \(\frac{5^{8}}{5^{2}}\) in your explanation.

Step-by-Step Solution

Answer

Using the Quotient Rule for Exponents, \(\frac{5^{8}}{5^{2}}\) simplifies to \(5^{6}\).

Step 1: Understanding the Quotient Rule for Exponents

The Quotient Rule for Exponents states that when dividing like bases, subtract the exponents. In general terms, for any non-zero number a, it's: \(a^{m}/a^{n} = a^{m-n}\), where m and n are any real numbers.

Step 2: Identify the Base and Exponents

For the expression \(\frac{5^{8}}{5^{2}}\), the base (a) is 5, the exponent in the numerator (m) is 8, and the exponent in the denominator (n) is 2.

Step 3: Apply the Quotient Rule

Subtract the denominator's exponent from the numerator's exponent: \(5^{8-2} = 5^{6}\)