Problem 8 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ \left(3^{4}\right)^{-1} $$

Step-by-Step Solution

Answer

The evaluation of the expression \((3^4)^{-1}\) without the use of a calculator gives \(\frac{1}{81}\) as our result.

Step 1: Apply the power of a power rule

This rule states that \((a^m)^n = a^{m \cdot n}\). Applying this rule to our expression, \((3^4)^{-1}\) becomes \(3^{4 \cdot -1}\).

Step 2: Determine the value of the exponent

By multiplying \(4 \cdot -1\), our exponent becomes \(-4\). Therefore, our expression is simplified to \(3^{-4}\).

Step 3: Apply the negative exponent rule

This rule states that \(a^{-n} = \frac{1}{a^n}\). Applying this rule to our expression, \(3^{-4}\) becomes \(\frac{1}{3^4}\).

Step 4: Evaluate the expression

As per the laws of exponents, \(3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\). Replacing 81 into the expression \(\frac{1}{3^4}\), we get \(\frac{1}{81}\).

Key Concepts

Understanding the Power of a Power Rule

When dealing with exponents, the power of a power rule simplifies the process of raising a power to another power. The rule states that when you have an expression of the form \((a^m)^n\), you can multiply the exponents together to find the new power of the base number, resulting in \(a^{m \cdot n}\). Think of this rule as an efficient shortcut for exponential growth.

For example, consider the expression \((3^4)^{-1}\). Here, the base number 3 is raised to the 4th power, and then that entire expression is raised to the negative first power. According to the power of a power rule, you multiply the exponents: 4 \(-1\) = -4, so the expression becomes \(3^{-4}\). This rule is fundamental in evaluating expressions with exponents because it simplifies the process into a single step, and saves you from cumbersome calculations.

Applying the Negative Exponent Rule

The negative exponent rule is one of the more intriguing concepts in mathematics that often confuses students. It states that any number a, raised to a negative exponent n, is equal to one divided by that number raised to the corresponding positive exponent: \(a^{-n} = \frac{1}{a^n}\).

In the given expression \(3^{-4}\), we apply the negative exponent rule to transform it into \(\frac{1}{3^4}\). The negative sign in the exponent essentially 'flips' the position of the number in a fraction, turning a base with a negative exponent into the denominator of a fraction with a positive exponent. It's a simple yet powerful rule that allows you to handle expressions that at first glance seem counterintuitive.

Leveraging the Laws of Exponents

The laws of exponents, or rules for handling expressions with exponents, include the previously discussed power of a power rule and negative exponent rule, among others. These laws allow us to perform operations on powers efficiently and thereby simplify expressions. One fundamental law is the multiplication of like bases: \(a^m\cdot a^n = a^{m+n}\). Another is the division of like bases: \(\frac{a^m}{a^n} = a^{m-n}\).

Using these laws helps in evaluating the final expression \(\frac{1}{3^4}\). Here, you would calculate the positive exponent by multiplying 3 by itself 4 times, resulting in 81. Therefore, the entire expression evaluates to \(\frac{1}{81}\). Understanding and applying the laws of exponents transform complex exponential expressions into basic arithmetic, making it simpler for students to solve these problems without the need for a calculator.