Problem 124 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result of \(\frac{5^{8}}{5^{2}}\) using the quotient rule is \(5^{6}\).

Step 1: Understanding Quotient Rule

Let's recall the Quotient Rule for exponents which states: \(a^{m}/a^{n} = a^{m-n}\), where 'a' is any non-zero number, 'm' and 'n' are any real numbers. So, if we are dividing two numbers where the base is identical, we simply subtract the exponent of the denominator from the exponent of the numerator.

Step 2: Applying Quotient Rule

Now apply the rule to the given expression \(\frac{5^{8}}{5^{2}}\). Here, our base is 5 and we have 8 as the exponent in the numerator and 2 as the exponent in the denominator.

Step 3: Subtracting Exponents

According to the Quotient Rule, we subtract the exponent of the denominator from the exponent of the numerator: \(5^{8-2}\).

Step 4: Solve the Exponent Subtraction

Calculate \(5^{8-2}\) to get \(5^{6}\).