Problem 101 - College Algebra

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 for exponents, \(\left(3^{2}\right)^{4} = 3^{8}\)

Step 1: Understand the power rule for exponents

The power rule for exponents states that for any number 'a', and integers 'm' and 'n', \((a^{m})^{n} = a^{mn}\). This means that when an exponential expression is raised to a power, the powers multiply.

Step 2: Apply the power rule to the given example

Apply the rule to the given example, \(\left(3^{2}\right)^{4} = 3^{(2*4)}\). This follows directly from the power rule.

Step 3: Calculate the value

Now compute the value of the expression: \(3^{(2*4)} = 3^{8}\).