Problem 125 - 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, the expression \( (3^2)^4 \) simplifies down to \( 3^8 \).

Step 1: Understanding the Power Rule

The Power Rule for Exponents states that for any numbers a, m and n, the term \( (a^m)^n \) is equal to \( a^{mn} \). The exponents m and n are multiplied together.

Step 2: Applying the Power Rule

Now, using the Power Rule, apply it to the given expression \( (3^2)^4 \). The base number is 3, the first exponent is 2 and the second exponent is 4. So, we multiply the exponents 2 and 4 together. This leads to \( 3^{2*4} \).

Step 3: Simplifying the Expression

After you multiply the exponents, the expression simplifies down to \( 3^8 \).