Problem 122 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Product rule for exponents states that \(a^{m} \cdot a^{n} = a^{m+n}\). So \(2^{3} \cdot 2^{5} = 2^{3+5} = 2^{8} = 256\.

Step 1: Introduction of the Rule: Product of Powers

The Product of Powers rule states that when multiplying two expressions with the same base, just add the exponents together. Formally it can be written as: \(a^{m} \cdot a^{n} = a^{m+n}\). In this exercise \(a = 2\), \(m = 3\), and \(n = 5\) so \(a^{m} \cdot a^{n}\) becomes \(2^{3} \cdot 2^{5}\).

Step 2: Application of the Rule

Applying the rule to example \(2^{3} \cdot 2^{5}\), the exponents (3 and 5) on base 2 should be added together, to give \(2^{3+5}\).

Step 3: Solving the Expression

Finally, the expression becomes \(2^{8}\). This can be calculated as 2 multiplied by itself 8 times, which results to 256.