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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 expression \(\left(3^{2}\right)^{4}\) simplifies to \(3^{8}\) using the power rule for exponents.

Step by step solution

01

Understand the power rule for exponents.

The power rule for exponents says that for any real number \(a\), and any integers \(m\) and \(n\), \( (a^m)^n = a^{m \cdot n} \). This means when an exponent is raised to another power, you multiply the exponents.
02

Apply the power rule to the given example.

Applying this rule to \(\left(3^{2}\right)^{4},\) we see that \(m = 2\) and \(n = 4\). So, we can multiply these two exponents. Substituting these values, the expression becomes \(3^{2 \cdot 4}\).
03

Simplify the Expression

Upon multiplying, the expression becomes \(3^{8}\). So, \(\left(3^{2}\right)^{4}=3^{8}\).

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Each statement applies to the division problem $$\frac{x^{3}+1}{x+1}$$ The purpose of writing \(x^{3}+1\) as \(x^{3}+0 x^{2}+0 x+1\) is to keep all like terms aligned.

Explain how to add polynomials.

Explain how to divide monomials. Give an example.

perform the indicated operations. $$\begin{aligned} &\left[\left(4 y^{2}-3 y+8\right)-\left(5 y^{2}+7 y-4\right)\right]-\left[\left(8 y^{2}+5 y-7\right)+\left(-10 y^{2}+4 y+3\right)\right] \end{aligned}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\left(6 x^{2} y-7 x y-4\right)-\left(6 x^{2} y+7 x y-4\right)=0$$
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