Problem 102 - College Algebra

Question

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

Step-by-Step Solution

Answer

The quotient of \(5^{8}\) and \(5^{2}\) is \(5^{6}\), following the quotient rule for exponents.

Step 1: State the Problem

Given the problem is to explain the quotient rule using the example \(\frac{5^{8}}{5^{2}}\).

Step 2: Describe the Quotient Rule

The quotient rule for exponents states that when you divide terms with the same base, you subtract the exponent of the divisor from the exponent of the dividend. This can be represented as \(a^{n}/a^{m} = a^{n - m}\).

Step 3: Apply the Quotient Rule

Applying this rule to the given example, the quotient of \(5^{8}\) and \(5^{2}\) can be expressed as \(5^{8 - 2}\).

Step 4: Simplify the Expression

After the subtraction in the exponent, the final expression becomes \(5^{6}\).