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Chapter 0: Problem 123

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

Short Answer

Expert verified
By using the power rule, \((3^2)^4\) simplifies to \(3^8\).

Step by step solution

01

Understand the Power Rule

The power rule for exponents states that, for any real numbers a, m, and n, when you have an exponent raised to another exponent, the exponents multiply together. This means \((a^n)^m = a^{n*m}\).
02

Identify Values

We have the expression \((3^2)^4\). Here, a = 3, n = 2, and m = 4.
03

Apply the Power Rule

Following the power rule, we multiply the exponents together: 2 * 4 = 8
04

Final Simplification

Substitute the base 'a' (which is 3) and the result of the multiplication of the exponents into the formula leading to \(3^{8}\).

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