Problem 124 - College Algebra Essentials

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, \(2^{3} \cdot 2^{5} = 2^{8}\) which equals 256.

Step 1: Understand the Product Rule

The product rule for exponents states that for any numbers a, m, and n, the multiplication of \(a^{m}\) and \(a^{n}\) can be simplified by adding the exponents. So, \(a^{m} \cdot a^{n} = a^{m+n}\). This rule can be logically explained by expanding the expressions. \(a^{m}\) means that 'a' is multiplied by itself 'm' times and \(a^{n}\) implies 'a' is multiplied by itself 'n' times. So when you multiply \(a^{m}\) and \(a^{n}\), you're multiplying 'a' by itself 'm + n' times.

Step 2: Apply the Product Rule

In this particular problem, we are multiplying \(2^{3}\) by \(2^{5}\), which has a base of '2' and exponents of '3' and '5'. Applying the product rule for exponents, we add the exponents. So, \(2^{3} \cdot 2^{5} = 2^{3+5}\) which simplifies to \(2^{8}\).

Step 3: Calculate the Final Result

Finally, to find the result of \(2^{8}\), multiply '2' by itself '8' times. This results in a final answer of 256.