Problem 125 - College Algebra Essentials

Question

Explain the power rule for exponents. Use \(\left(3^{2}\right)^{4}\) in your explanation.

Step-by-Step Solution

Answer

Using the power rule, \(\left(3^{2}\right)^{4}\) simplifies to \(3^{8}\).

Step 1: Understand the Power Rule

The power rule states that \((a^{m})^{n}=a^{m*n}\), where \(a\) is the base, \(m\) is the exponent of the base, and \(n\) is the outer exponent.

Step 2: Apply the Power Rule to Our Example

Using the power rule for the expression \(\left(3^{2}\right)^{4}\), 3 is the base (\(a = 3\)), 2 is the exponent of the base (\(m = 2\)), and 4 is the outer exponent (\(n = 4\)). According to the power rule, the equation simplifies to \(3^{2*4}\).

Step 3: Simplify the Expression

Now, simplify the expression by calculating \(2*4\) in the exponent to get \(3^{8}\).