Problem 100 - College Algebra

Question

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

Step-by-Step Solution

Answer

The result of \(2^{3} \cdot 2^{5}\) according to the product rule for exponents is \(2^{8}\)

Step 1: Understand the Product Rule for Exponents

The product rule for exponents states that, for any numbers \(a\), \(m\), and \(n\), \(a^{m} \cdot a^{n} = a^{m+n}\). This means if you are multiplying two expressions that have the same base, you can simply add the exponents.

Step 2: Apply the Product Rule to Given Problem

In the given problem, we have \(2^{3} \cdot 2^{5}\). According to the product rule, this can be simplified by adding the exponents since the bases (which are 2 in both terms) are the same. So it can be rewritten as: \(2^{(3+5)}\)

Step 3: Calculate

Perform the addition in the exponent: \(2^{8}\)