/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 Simplify each of the numerical e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 38

Simplify each of the numerical expressions. $$ 5(-1)^{3}-(-3)^{3} $$

Short Answer

Expert verified
The simplified expression is 22.

Step by step solution

01

Simplify the Exponentiated Terms

First, calculate the value of \((-1)^3\) and \((-3)^3\).\[(-1)^3 = -1 \times -1 \times -1 = -1\]\[(-3)^3 = -3 \times -3 \times -3 = -27\]
02

Apply the Negative Sign

Now, account for the negative sign in front of the \((-3)^3\) term.Since there is a negative sign in front of \((-3)^3\), we have:\(-(-3)^3 = -(-27) = 27\).
03

Substitute Back into the Expression

Substitute the simplified values back into the original expression:\[5(-1)^3 - (-3)^3 = 5(-1) + 27\]
04

Calculate the Result

Now, multiply the 5 and \(-1\): \[5(-1) = -5\] Then perform the addition:\(-5 + 27 = 22\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Exponentiation
Exponentiation is the mathematical operation where a number, known as the base, is multiplied by itself a specified number of times, called the exponent. For example, in the expression \((-1)^3\),
  • the base is \(-1\)
  • the exponent is 3
This means that \((-1)\) should be multiplied by itself 3 times, resulting in: \[ (-1) \times (-1) \times (-1) = -1 \]When dealing with odd exponents and negative bases, the result retains the negative sign. In contrast, negative bases raised to an even exponent will result in a positive product.
  • For example, \((-2)^2 = 4\)
  • Whereas \((-2)^3 = -8\)
Understanding this behavior helps in correctly simplifying expressions involving exponents.
Simplifying Expressions
Simplifying expressions often involves breaking them down into more manageable parts, making calculations more straightforward. In our example, we have the expression \(5(-1)^3 - (-3)^3\).
  • We start by addressing exponentiation, as this is fundamental before other operations.
  • Once exponentiated components are resolved, move to arithmetic operations such as addition or subtraction.
In our process:
  • \((-1)^3\) was simplified to \(-1\)
  • \((-3)^3\) became \(-27\)
Next, handling any negative signs correctly is crucial. Negative signs in front of exponential terms can change how they are combined. After simplifying the exponents, substitute them back into the original expression. Finish by performing the straightforward arithmetic to achieve the final simplification like this:
  • Substitution gives \(5(-1) + 27\)
  • Finally simplifies to \(-5 + 27 = 22\)
These steps showcase the importance of a systematic approach in simplifying numerical expressions.
Negative Numbers
Negative numbers, those less than zero, exhibit unique properties that can affect outcomes in arithmetic operations. When working with expressions that include negative numbers, special attention is needed:
  • Multiplying two negative numbers results in a positive number, e.g., \((-2) \times (-3) = 6\)
  • Multiplying a negative number by a positive number keeps the result negative, e.g., \((-2) \times 3 = -6\)
In the exercise, negative numbers were primarily present in the bases of exponents. Notice how influencing factors like the exponent also determine the overall sign:
  • \((-1)^3\) maintained its negative value, while
  • \(-(-3)^3\) inverted the sign to positive due to the initial subtraction sign.
Managing sign changes is a crucial skill when simplifying expressions involving negative numbers. It's often helpful to visualize the process or write down intermediary sign results to avoid mistakes.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 3(x+y)-2(x-y) $$

Answer the question with an algebraic expression. Tina has \(c\) cents, which is all in quarters. How many quarters does she have?

Answer the question with an algebraic expression. The perimeter of a square is \(i\) inches. What is the perimeter expressed in feet?

Evaluate the algebraic expressions for the given values of the variables. $$ 4 x^{2}-y^{2}, x=2 \text { and } y=-2 $$

Answer the question with an algebraic expression. Mila's monthly salary is \(d\) dollars. What is her annual salary?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.