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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
Using the power rule for exponents, the expression \( \left(3^{2}\right)^{4}\) simplifies to \(3^{8}\), and evaluating this gives 6,561.

Step by step solution

01

Understanding the power rule

The power rule for exponents states that when powers are raised to another power, the exponents are multiplied. This rule is represented by the formula \((a^{m})^{n} = a^{m*n}\), where \(a\) is the base number and \(m\) and \(n\) are the exponents.
02

Applying the power rule

Applying this rule to the expression \( \left(3^{2}\right)^{4}\), \(m\) equals 2 (the first exponent), \(n\) is 4 (the second exponent) and the base \(a\) is 3. So according to the rule, this can be rewritten as \(3^{2*4}\).
03

Solving the problem

Now, you can perform the multiplication in the exponent. \(2*4=8\), so the expression simplifies to \(3^{8}\).
04

Final evaluation

Once you made sure your exponent is correct, complete the operation. \(3^{8}\) equals 6,561.

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