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Chapter 2: Problem 54

Simplify. $$(-1)^{3}$$

Short Answer

Expert verified
The simplified form of \((-1)^3\) is \(-1\).

Step by step solution

01

- Understand the Problem

Recognize that you need to simplify the expression \((-1)^3\). This involves using the properties of exponents.
02

- Apply the Exponent

Recall that raising a number to an exponent means multiplying the number by itself as many times as the exponent indicates. For example, \(a^3 = a \times a \times a\).
03

- Substitute and Calculate

Substitute \(-1\) into the exponent equation: \((-1)^3 = (-1) \times (-1) \times (-1)\).
04

- Multiply Step by Step

First, compute \((-1) \times (-1) = 1\). Then multiply the result by \(-1\): \(1 \times (-1) = -1\).
05

- State the Final Answer

After multiplying, the final simplified value of \((-1)^3\) is \(-1\).

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Key Concepts

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

Exponents Basics
Exponents are a fundamental concept in mathematics that makes it easier to write repeated multiplication. When you see a number written with an exponent, like this: \(a^n\), it means that the base \(a\) is multiplied by itself \(n\) times. For example, in \(2^3\), the number 2 (the base) is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\).

Here are some common properties of exponents that are useful to know:
  • Any number raised to the power of 1 is the number itself, i.e., \(a^1 = a\).
  • Any non-zero number raised to the power of 0 is 1, i.e., \(a^0 = 1\).
  • For multiplication of like bases with exponents, you add the exponents: \(a^m \times a^n = a^{m+n}\).

Remember, understanding these properties can simplify many mathematical operations involving exponents.
Negative Numbers
Negative numbers are numbers less than zero. They are often represented with a minus sign, such as -1, -2, -3, and so forth. Working with negative numbers follows some specific rules:
  • The product of two negative numbers is positive: \((-a) \times (-b) = ab\).
  • The product of a negative number and a positive number is negative: \((-a) \times b = -ab\).
  • Adding two negative numbers results in a more negative number, e.g., \(-2 + (-3) = -5\).

These rules are essential when simplifying expressions with exponents that involve negative numbers, as in our exercise \((-1)^3\).
Multiplication Rules
Multiplication is key when working with exponents. Here鈥檚 what you need to keep in mind:
  • Multiplying two positive numbers results in a positive product: \(a \times b = ab\).
  • Multiplying a positive number by a negative number results in a negative product: \(a \times (-b) = -ab\).
  • Multiplying two negative numbers results in a positive product: \((-a) \times (-b) = ab\).

So, during our simplification of \((-1)^3\), we multiply \((-1) \times (-1) = 1\) first, and then \(1 \times (-1) = -1\).
Following these rules ensures correct outcomes in problems involving multiplication of negative numbers or numbers with exponents.

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Most popular questions from this chapter

When a severe winter storm moved through Albany, New York, the change in temperature between 4: 00 P.M. and 7: 00 P.M. was \(-27^{\circ} \mathrm{F}\). What was the average hourly change in temperature?

a. What number must be multiplied by \(-5\) to obtain \(-35 ?\) b. What number must be multiplied by \(-5\) to obtain \(35 ?\)

Assume \(a>0\) (this means that \(a\) is positive) and \(b<0\) (this means that \(b\) is negative). Find the sign of each expression. $$a \cdot|b|$$

Translate the English phrase to a mathematical expression. Then simplify. The quotient of 46 and \(-23\)

Rank the numbers from least to greatest. If \(d\) represents a negative number, then what is the sign of \(-(-d) ?\)
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