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Chapter 0: Problem 2

In Exercises \(1-28\), compute the numbers. $$ (-2)^{3} $$

Short Answer

Expert verified
-8

Step by step solution

01

- Identify the Base and Exponent

The base number is -2, and the exponent is 3.
02

- Write the Expression in Expanded Form

To compute \((-2)^3\), write it as \(-2 \times -2 \times -2\).
03

- Multiply the First Two Terms

Multiply the first two terms: \(-2) \times (-2) = 4\).
04

- Multiply the Result by the Last Term

Multiply the result \(4\) by the last term \(-2\): \(4 \times (-2) = -8\).

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Key Concepts

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

Understanding Base and Exponent
In the expression \((-2)^3\), we have two main components: the base and the exponent. The base is the number that we repeatedly multiply. Here, the base is -2.
The exponent tells us how many times the base is used as a factor. In this case, the exponent is 3, which means we will multiply -2 three times.
So, when dealing with exponents, always identify two things:
  • Base: The number being multiplied repeatedly.
  • Exponent: The number of times the base is multiplied by itself.
Expanded Form
Writing an expression in expanded form means breaking it down into its multiplication components. For \((-2)^3\), we write:\[\begin\{align*\}-2 \times -2 \times -2\end\{align\}\]Here, -2 is multiplied by itself twice more. Doing this makes the calculation easier.
Each step deals with two numbers at a time, simplifying the process.
Expanded forms help visualize and compute values more quickly, especially when combined with understanding base and exponent.
Multiplication of Integers
Integer multiplication follows these basic rules:
  • Positive x Positive: The result is positive.
  • Negative x Negative: The result is positive.
  • Positive x Negative: The result is negative.

In our example, \((-2)\times(-2) = 4\), because multiplying two negative integers gives a positive result.
Next, when we multiply 4 by -2, we get:\[4\times-2 = -8\]This final multiplication involves a positive and a negative integer, hence the result is negative.
Remembering these rules of integer multiplication makes solving these problems much easier!

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Most popular questions from this chapter

Use the laws of exponents to compute the numbers. \((125 \cdot 27)^{1 / 3}\)

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