Problem 126 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression 💲\frac{5^{8}}{5^{2}}💲, using the quotient rule for exponents, is 💲5^{6}💲.

Step 1: Understanding the Quotient Rule for Exponents

The quotient rule for exponents states that when dividing two power expressions with the same base, subtract the exponent of the denominator from that of the numerator. This can be written as: \(a^{m}/a^{n} = a^{m-n}\), where \(m\) and \(n\) are exponents and \(a\) is the common base.

Step 2: Using the Quotient Rule on the Given Expression

The given expression is \(\frac{5^{8}}{5^{2}}\). Here, the base \(a\) is 5, the exponent in the numerator \(m\) is 8, and the exponent in the denominator \(n\) is 2. Applying the quotient rule gives: \(\frac{5^{8}}{5^{2}} = 5^{8-2}\).

Step 3: Final Calculation

Having simplified the expression using the quotient rule, we can now proceed to calculate the result: \(5^{8-2} = 5^{6}\).