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Chapter 1: Problem 3

Perform each operation. $$4 \cdot 4 \cdot 4$$

Short Answer

Expert verified
The result of the multiplication \(4 \cdot 4 \cdot 4\) is 64.

Step by step solution

01

Identifying the multiplication operation

The exercise involves multiplying 4 by itself three times. In mathematical terms, this is also known as cubing a number. Here 4 is cubed.
02

Performing the multiplication operation

First, multiply the first two 4's together, the result is 16. Then multiply 16 by the final 4. The result of this operation is 64.

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Key Concepts

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

Understanding Multiplication in Exponents
Multiplication is a fundamental arithmetic operation where we combine equal groups. In the context of exponents, multiplication becomes a convenient tool to simplify repeated addition. The original exercise involving multiplying 4 by itself three times, is an excellent example of this. The numbers 4, 4, and 4 multiply together sequentially, and instead of adding 4 three times, which looks like this: 4 + 4 + 4, multiplication allows us to compute it faster with repeated multiplication.
  • Start by multiplying the first two 4's: \( 4 \times 4 = 16 \).
  • Then, multiply the result by the last 4: \( 16 \times 4 = 64 \).
This process demonstrates how multiplication simplifies calculations. Each step depends on accurately multiplying preceding results until reaching a final answer.
Cubing Numbers and Its Significance
Cubing refers to raising a number to the power of three. This means multiplying the number by itself twice. Taking 4 cubed, which is mathematically represented as \(4^3\), produces the result the exercise underlines. When you cube a number, it scales up its size significantly because you're essentially multiplying three instances of it.
  • The process is: 4 multiplied by 4 gives 16.
  • Then, multiplying 16 by another 4 gives 64.
Such repetition of multiplication shows how numbers expand in size when repeatedly multiplied by themselves. Cubing is relevant in everyday applications like determining volume, where all three dimensions (length, width, height) are the same, making it a powerful concept to grasp in mathematics.
The Role of Arithmetic Operations
Arithmetic operations include addition, subtraction, multiplication, and division. Each plays a unique role in calculated processes. This exercise focused on multiplication鈥攐ne of the key operations, particularly for exponents. Multiplication is useful in situations where you need to quickly scale a number by itself multiple times, reducing potentially tedious repeated addition or substraction. Understanding these operations:
  • Addition: Combining quantities, whereas multiplication handles repeated addition efficiently.
  • Subtraction: Removing or decreasing quantities, not directly applicable here.
  • Division: Splitting or distributing quantities, the inverse of multiplication.
In essence, grasping each operation's purpose is vital for solving tasks involving exponents. It helps see connections between arithmetic principles and everyday mathematical challenges.

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Most popular questions from this chapter

A high-efficiency home heating system can be installed for \(\$ 4,170,\) with an average monthly heating bill of \(\$ 57.50 .\) A regular furnace can be installed for \(\$ 1,730,\) but monthly heating bills average \(\$ 107.75 .\) After three years, which system has cost more altogether?

Which property of real numbers justifies each statement? $$x(y+z)=(y+z) x$$

Suppose there were no numbers other than the odd integers. \(\cdot\) Would the closure property for addition still be true? \(\cdot\) Would the closure property for multiplication still be true? \(\cdot\) Would there still be an identity for addition? \(\cdot\) Would there still be an identity for multiplication?

Two carpenters bid on a home remodeling project. The first bids \(\$ 9,350\) for the entire job. The second will work for \(\$ 27.50\) per hour, plus \(\$ 4,500\) for materials. He estimates that the job will take 150 hours. Which carpenter has the lower bid?

Distinguish between factor and term.
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