/*! 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 12 Find the product. $$-(-1)^{5}$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 12

Find the product. $$-(-1)^{5}$$

Short Answer

Expert verified
The product is 1.

Step by step solution

01

Exponentiation

Perform the exponentiation operation inside the parentheses. Any number to the odd power maintains it sign. Hence, \( (-1)^{5} = -1 \)
02

Negation operation

The negation of a number changes the sign of the number. So, the negative sign in front of the parentheses changes the sign of the result we obtained in the previous step. In other words, we take the negation of -1, which becomes positive: \(-(-1) = 1 \).

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 a fundamental mathematical operation involving two numbers, the base, and the exponent. It is written as \( a^n \), where \( a \) is the base and \( n \) is the exponent. This expression represents the number \( a \) multiplied by itself \( n \) times. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
In the given exercise, the base is -1 and the exponent is 5, an odd number.
  • When the base is negative and the exponent is an odd number, the resulting product keeps the negative sign. This is because multiplying an odd number of negative values results in a negative product. Thus, \( (-1)^5 = -1 \).
  • Remember that if the exponent were an even number, the product would be positive due to the pairing of negative signs, resulting in a positive product.
Understanding the behavior of exponents with negative bases is key to mastering exponentiation.
Negation
Negation in mathematics refers to changing the sign of a number. If the number is positive, its negation is negative, and vice versa.
When using the '-' sign in front of a number or an expression, it indicates negation, which essentially involves multiplying by -1.
  • In our exercise, after solving \((-1)^5 = -1\), we are left with the expression \(-(-1)\).
  • Applying the negation operation to -1 changes its sign to positive 1. So, \(-(-1) = 1\), as negating a negative effectively "undoes" the negative, turning it into positive.
This operation is vital for solving problems involving integers, especially when dealing with expressions with multiple layers of negative signs.
Odd Exponents
Odd exponents are exponents that are odd integers such as 1, 3, 5, 7, and so on. They fundamentally affect the sign of a product when dealing with negative bases.
An odd exponent on a negative base value retains the negative sign of the base.
  • Consider a negative base like -2 raised to an odd exponent, like 3: \((-2)^3 = -2 \times -2 \times -2 = -8\).
  • A key insight is that each pair of negative numbers results in a positive product, but since the exponent is odd, there is one unpaired negative, ensuring the final result is negative.
This is a crucial pattern to keep in mind when performing calculations with integer operations, as it governs the final sign of the expression's result.

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

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ 9 x^{3}-4 x^{3}-2 $$

BUYING JEANS You have 58 dollar, and you want to buy a pair of jeans and a \(\$ 20\) T-shirt. There is a \(6 \%\) sales tax. If \(x\) represents the cost of the jeans, then the following inequality is a model that shows how much you can spend on the jeans. $$ x+20+0.06(x+20) \leq 58 $$ If the jeans cost \(\$ 35,\) can you buy both the T-shirt and the jeans?

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ -r(r-9) $$

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ -4(y+2)-6 y $$

LEAVING A TIP In Exercises \(83-85\), use the following information. You and a friend decide to leave a \(15 \%\) tip for restaurant service. You compute the tip, \(T,\) as \(T=0.15 C,\) where \(C\) represents the cost of the meal. Your friend claims that an easier way to mentally compute the tip is to calculate \(10 \%\) of the cost of the meal plus one half of \(10 \%\) of the cost of the meal. Will both methods give the same results? Explain.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

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.