Problem 18 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$\frac{3^{8}}{3^{4}}$$

Step-by-Step Solution

Answer

The evaluation of the given expression is 81.

Step 1: Apply the properties of exponents

We can use the property \(a^{m} / a^{n} = a^{m-n}\) which states when dividing like bases, you should subtract the exponents. So the given expression \(\frac{3^{8}}{3^{4}}\) becomes \(3^{8-4}\).

Step 2: Simplify the exponent

Simplifying the exponent gives \(3^{4}\).

Step 3: Evaluation of the exponential expression

By evaluating this, we know that \(3^{4}\) equals 81.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial in evaluating exponential expressions. Exponents illustrate how many times a number, known as the base, is multiplied by itself. One such property is the Quotient of Powers Property, which states that when you divide two exponents with the same base, you can subtract the exponents. In mathematical form, it's written as \( a^{m} \div a^{n} = a^{m-n} \). This rule was applied in the step-by-step solution when \( \frac{3^{8}}{3^{4}} \) was simplified to \( 3^{8-4} \).
Other properties include the Product of Powers Property \( a^{m} \times a^{n} = a^{m+n} \) and the Power of a Power Property \( (a^{m})^{n} = a^{m \times n} \). Recognizing and properly applying these rules can significantly simplify the process of solving exponential problems.

Simplifying Exponents

When simplifying exponents, the goal is to make the expression as elementary as possible. This involves using the properties of exponents. After identifying that the bases are the same, exponents can often be combined or reduced. For example, in the given problem, we consolidated \( 3^{8} \div 3^{4} \) into \( 3^{4} \) by subtracting the exponents. It's a straightforward subtraction since the bases are identical.
It is also important to be aware of specific 'special exponents' such as \( a^{0} = 1 \) and \( a^{1} = a \). Memorizing these can help avoid common mistakes and make simplification faster.

Exponential Expression Evaluation

Evaluating exponential expressions involves performing the necessary calculations to find the value of the expression. After simplifying the expression using the properties of exponents, the next step is computation. With \( 3^{4} \) from the aforementioned exercise, the evaluation step is essentially calculating \( 3 \times 3 \times 3 \times 3 \), which equals 81. When evaluating, it's often helpful to break down the expression into familiar multiplication facts or to use a calculator for larger exponents to ensure accuracy.

Algebraic Problem-Solving

Algebraic problem-solving involves identifying the problem, strategizing a solution method, and carrying out the necessary steps in an orderly manner. In the case of exponential expressions, the strategy includes applying properties of exponents and simplification at the outset. Once the expression is simplified, evaluation follows through computation. Finally, it's essential to review your work to confirm that all algebraic rules have been correctly applied and that the evaluation is precise. In the exercise provided, all three steps—Application of Properties, Simplification, and Evaluation—were executed meticulously to arrive at the correct answer, demonstrating effective algebraic problem-solving.