Problem 16 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$\left(3^{3}\right)^{2}$$

Step-by-Step Solution

Answer

The given exponential expression \((3^3)^2\) evaluates to 729.

Step 1: Understand the problem

The aim is to evaluate the exponential expression \((3^3)^2\). According to the law of exponents, the powers need to be multiplied when a power raises to another power. In this case, the base 3 is raised to the power 3, and then the result is raised further to the power of 2.

Step 2: Apply the law of exponents

Set up the identification \(a = 3^3\), \(m = 3\), \(n = 2\). Thus, \((a^m)^n = a^{m \times n}\) implies \((3^3)^2 = 3^{3 \times 2}\). This simplifies to \(3^6\).\n\nThe base here is 3, and it is being raised to the power of 6. The next step is to simplify this expression.

Step 3: Evaluate the expression

Evaluate the expression \(3^6\). This calculation results in the number 729.

Key Concepts

Law of Exponents

The law of exponents is crucial to understanding how to manipulate and simplify expressions involving powers. An exponential expression like \((a^m)^n\) can be tricky to evaluate if you don't know how to apply the law of exponents effectively. According to this law, when you have a power raised to another power—like this expression—you multiply the exponents. So \((a^m)^n = a^{m \times n}\).

This law is powerful because it allows us to combine multiplications of the same base into a single expression, thus simplifying our calculations considerably. It's like saying if you have a bunch of small stacks of the same size blocks, and you stack those stacks on top of each other, you end up with one big stack. The height of that big stack is just the total number of blocks, or in our case, the base number raised to the power of all the stacked exponents.

Evaluate Exponents

Evaluating exponents is the process of finding out what a number raised to a certain power is. For example, when you see \(3^6\), you are looking at the base number 3 raised to the power of 6. To evaluate exponents, we multiply the base number by itself as many times as indicated by the exponent, so \(3^6\) is 3 multiplied by itself a total of five more times, because we always start with one occurrence of the base: 3 x 3 x 3 x 3 x 3 x 3 = 729.

When evaluating exponents, it is important to remember that the exponent indicates how many times the base appears as a factor, not how many multiplications you perform. That is why, even though 6 seems to suggest six multiplications, we actually have 3 as a factor six times.

Simplifying Exponential Expressions

Simplifying exponential expressions is about making them as straightforward as possible, often times by using the law of exponents. Starting with expressions like \((3^3)^2\), we first look to transform it to a base with only one exponent using the rules we've discussed. Once we have \(3^6\), we then evaluate the exponent, finding the simplest form of the expression—729, in our case.

When simplifying, it's important to keep track of our bases and ensure that we only apply the law of exponents when the bases are the same. Additionally, there are shortcuts that can be used for common powers, such as squaring a number (raising to the power of 2) or cubing a number (raising to the power of 3), to make simplification faster. The overarching goal is to reduce the expression to numbers or simpler expressions that provide clarity and reduce complexity.