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Chapter 5: Problem 106

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

Short Answer

Expert verified
The Power Rule for exponents involves multiplying the exponents together, hence applying this rule to \(\left(3^{2}\right)^{4}\) results in \(3^{8}\).

Step by step solution

01

Understanding Exponents

An exponent refers to the number of times a number is multiplied by itself. For example, in \(3^{2}\), the base number 3 is multiplied by itself 2 times \(3 * 3\).
02

Power Rule

The Power Rule for exponents states that \( (a^{m})^{n} = a^{m*n} \). That means when you're raising a power by another power, you multiply the exponents. The rule is applicable on every real number a (except 0) and for all integers m and n.
03

Applying the Power Rule to \(\left(3^{2}\right)^{4} \)

Applying the Power Rule to \(\left(3^{2}\right)^{4}\) would result in multiplying the exponents by each other which gives: \(\left(3^{2}\right)^{4} = 3^{2*4} = 3^{8}\).

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