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Chapter 1: Problem 17

Find the value of each exponential expression. $$ 3^{4} $$

Short Answer

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Step by step solution

01

Understand the Exponential Expression

The expression is an exponential expression where the base is 3 and the exponent is 4. It is written as \( 3^{4} \).
02

Rewrite the Exponential Expression in Multiplicative Form

An exponential expression such as \( 3^{4} \) means multiplying the base, 3, by itself as many times as the exponent indicates. So, \( 3^{4} \) can be rewritten as \( 3 \times 3 \times 3 \times 3 \).
03

Perform the Multiplication

Now multiply the numbers together step-by-step:1. \( 3 \times 3 = 9 \)2. \( 9 \times 3 = 27 \)3. \( 27 \times 3 = 81 \)So, \( 3^{4} = 81 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base and Exponent
In an exponential expression like \(3^{4}\), there are two key components: the base and the exponent. The base is the number that is being multiplied. In this case, the base is \(3\). The exponent tells us how many times the base is used as a factor. Here, the exponent is \(4\).

Think of the base as the main number and the exponent as instructions for how to use that number. So, \(3^{4}\) tells us to multiply \(3\) by itself four times.

Understanding this helps in breaking down and solving exponential expressions with ease.
Multiplicative Form
To simplify an exponential expression, you can rewrite it in multiplicative form. This means expressing the exponential value as a product of the base multiplied by itself the number of times indicated by the exponent.

For example, \(3^{4}\) can be rewritten as \(3 \times 3 \times 3 \times 3\).

By breaking it down this way, it becomes simpler to handle and perform the multiplication. This step is crucial because it turns a potentially complex operation into a series of simpler multiplications.

Changing the form helps you see exactly what you are working with and makes solving easier.
Step-by-Step Multiplication
Once you have rewritten the exponential expression in its multiplicative form, solving it becomes a matter of simple multiplication.

For \(3^{4}\), here鈥檚 how to do it step-by-step:
  • First, multiply the base by itself: \(3 \times 3 = 9\).
  • Next, take the result and multiply it by the base again: \(9 \times 3 = 27\).
  • Finally, multiply by the base one last time: \(27 \times 3 = 81\).
So, \(3^{4} = 81\).

Following these steps carefully ensures that you get the correct result and helps you understand the process better. Each small multiplication brings you closer to the final answer, making it manageable.

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Most popular questions from this chapter

Solve each problem. Bruce Buit also enjoys playing Triominoes. On five successive turns, his scores were \(-13,15,-12,24,\) and \(14 .\) What was his total score for the five turns?

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