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Chapter 2: Problem 32

Evaluate. $$4^{3}$$

Short Answer

Expert verified
\(4^3 = 64\)

Step by step solution

01

Understand the exponentiation operation

For the expression \(a^n\), a is called the base, and n is called the exponent or power. In this exercise, the base is 4 and the exponent is 3.
02

Multiply the base by itself for the given number of times

Here, we have to multiply the base 4 by itself 3 times. This means: \(4 \times 4 \times 4\).
03

Calculate the result

Now, let's multiply the numbers to get the final result: \(4 \times 4 = 16\) and \(16 \times 4 = 64\). So, \(4^3 = 64\).

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Key Concepts

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

Base and Exponent
When we talk about **exponentiation**, we often mention two crucial components: the base and the exponent. These terms help us define how numbers are raised to powers, a fundamental concept in mathematics. Let's break down what each term means.
  • The base is the number that you are going to multiply by itself. In our example, the base is 4.
  • The exponent, also known as the power, tells you how many times to use the base in a multiplication. In this instance, the exponent is 3.
When you see an expression like \(4^3\), you're looking at a compact way to show that the base (4) is being multiplied by itself as many times as the exponent (3) dictates. So, it's really saying: 4 multiplied by 4 multiplied by 4.
Understanding the base and exponent is the first step to easily solving any exponentiation problem.
Math Problem-Solving
Math problem-solving is all about breaking down problems into smaller, more manageable steps. This makes it easier to address each part of the problem. In our case of evaluating \(4^3\), following a structured methodology greatly helps.
  • Understand the problem: Recognize that \(4^3\) involves multiplying the base, 4, by itself three times.
  • Solve step-by-step: Begin by calculating the simpler parts, such as \(4 \times 4\).
  • Keep going until you finish the problem: After getting the result for \(4 \times 4\), multiply the result by the base again to complete the equation.
By taking these steps, you make the problem easier to handle. Don't hesitate to break complex problems down and solve them one piece at a time. This technique can not only help with exponentiation problems but can be applied to all types of math challenges. It empowers you to solve problems confidently and accurately.
Powers of Numbers
The powers of numbers are a crucial mathematical concept that expresses repeated multiplication. They are often seen in algebra, geometry, calculus, and many other areas.
  • A power tells you how many times to use the number in a multiplication.
  • Each "power" increases the challenge slightly but follows the same principle of repeated multiplication.
  • The term "squared" is used for an exponent of 2, and "cubed" is used for an exponent of 3. For example, \(4^2\) ("4 squared") equals \(4 \times 4\), and \(4^3\) ("4 cubed") equals \(4 \times 4 \times 4\).
Understanding powers is essential because it simplifies expression and calculation of large numbers. Instead of having to write out and multiply numbers multiple times, knowing the power makes the process straightforward. Additionally, the powers of numbers can lead to fascinating patterns and shortcuts in math that you can use to your advantage.

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Most popular questions from this chapter

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