Problem 124 - College Algebra

Question

Explain the product rule for exponents. Use \(2^{3} \cdot 2^{5}\) in your explanation.

Step-by-Step Solution

Answer

Using the product rule for exponents, which states that for any numbers a, n, and m, the equation \(a^{n} \cdot a^{m} = a^{n+m}\) is true, we get \(2^{3} \cdot 2^{5} = 2^{8}\).

Step 1: Understanding the Product Rule for Exponents

The product rule for exponents states that for any numbers a, n, and m, the equation \(a^{n} \cdot a^{m} = a^{n+m}\) holds true. This means that when multiplying two powers (like \(a^{n}\) and \(a^{m}\)) that share the same base (in this case a), their exponents can be added together.

Step 2: Applying the Product Rule

Now apply the product rule to the specific example given, \(2^{3} \cdot 2^{5}\). Here, the base 'a' is 2, 'n' is 3 and 'm' is 5. According to the product rule, add these two exponents together: \(2^{3} \cdot 2^{5} = 2^{3+5}\).

Step 3: Simplifying the Equation

The final step is to simplify the equation \(2^{3+5}\) by performing the operation in the exponent. This gives: \(2^{8}\). Thus, \(2^{3} \cdot 2^{5} = 2^{8}\).