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Chapter 5: Problem 27

Use properties of exponents to simplify each expression. First express the answer in exponential form. Then evaluate the expression. \(3^{-3} \cdot 3\)

Short Answer

Expert verified
The simplified form of the expression is \(3^2\) and the evaluated expression equals 9.

Step by step solution

01

Recall Exponents Properties

Before starting the problem, remember these key properties of exponents: 1) Any number raised to the exponent of -n is equal to 1 divided by the number raised to the n power (In other words, \(a^{-n} = 1/a^n\)). 2) When multiplying numbers with the same base, the exponents are added (\(a^n * a^m = a^{n+m}\)). Using these rules, the given expression will be simplified.
02

Apply the Negative Exponent Rule

From the given expression \(3^{-3} \cdot 3\), the negative exponent rule is applied to \(3^{-3}\). So, \(3^{-3}\) can be rewritten as \(\frac{1}{3^3}\). Therefore, the expression becomes \(\frac{1}{3^3} \cdot 3\)
03

Apply the Property of Exponents for Multiplication

Rewrite the multiplication \(\frac{1}{3^3} \cdot 3\) as \(\frac{1}{3^3} \cdot 3^1\). Now, apply the property of exponents that states when multiplying numbers with the same base, the exponents are added. Because the multiplication is happening over a fraction, it will be subtracted. So, the expression becomes \(3^{3-1}\).
04

Simplify the Expression

Now evaluate the expression \(3^{3-1}\) which simplifies to \(3^2\). Expressing this as a number by evaluating \(3^2\), the result is 9

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