/*! 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 3 Evaluate each exponential expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 3

Evaluate each exponential expression in Exercises 1鈥�22. $$ (-2)^{6} $$

Short Answer

Expert verified
The result of the expression \(-2^6\) is 64

Step by step solution

01

Understand the Exponentiation Rule

An exponent refers to the number of times a number is multiplied by itself. For example, \(a^n\) means a is multiplied by itself n times. If n is 6, it means a is multiplied by itself 6 times.
02

Applying the Exponent to -2

In the given expression \(-2^6\), -2 is the base while 6 is the exponent. This implies that -2 is multiplied by itself 6 times.
03

Calculate the Expression

Start multiplying. Note that a negative number multiplied by a negative number gives a positive result.\n\((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2)= 64\)

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 Rule
The exponentiation rule is a fundamental concept in mathematics that allows us to simplify how we express repeated multiplication of the same number. When you see any expression in the form of \(a^n\), it's telling us that the base \(a\) is multiplied by itself \(n\) times. For example, \(a^3 = a \times a \times a\). This rule helps us understand and calculate powers more efficiently.
  • The base is the number that is being multiplied.
  • The exponent shows how many times the base is multiplied by itself.
  • If the exponent is positive, like 6 in \((-2)^6\), we keep multiplying.
Understanding and applying this rule correctly ensures accurate calculations, which is crucial, especially when dealing with complex equations. It abstracts repetitive tasks into a simple notation, making mathematical expressions easier to handle.
Negative Bases in Exponents
Handling negative bases in exponents requires a careful approach. When using negative numbers as bases with exponential expressions, the outcome can differ based on whether the exponent is odd or even.Consider the expression \((-2)^6\). Here, -2 is the base, which means:
  • If the exponent is even (like 6 in this case), the result becomes positive. This occurs because the pairs of negative numbers multiply to produce positive results, such as \((-2) \times (-2) = 4\).
  • If the exponent is odd, the result remains negative as there will be an unpaired negative factor.
In our example, \((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2)\) ultimately resolves to a positive 64, because the six negative factors pair up and cancel out the negativity.
Multiplying Negative Numbers
Multiplying negative numbers is a key aspect of understanding exponential expressions with negative bases. Specific rules apply that must be memorized to accurately solve problems:
  • When you multiply two negative numbers, the product is positive: \((-a) \times (-b) = ab\). This happens because the negatives cancel each other out.
  • If you multiply a negative number by a positive number, the result is negative: \((-a) \times b = -ab\).
  • The repeated product of an even number of negative numbers results in a positive outcome.
  • An odd number of negative numbers multiplied together results in a negative product.
Practicing multiplication with negative numbers helps reinforce these rules, making complex expressions more manageable. In exponential expressions like \((-2)^6\), knowing that pairs of negative factors produce positive results provides clarity in computations.

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

a. Simplify: \(21 x+10 x\) b. Simplify: \(21 \sqrt{2}+10 \sqrt{2}\)

Describe what it means to raise a number to a power. In your description, include a discussion of the difference between \(-5^{2}\) and \((-5)^{2}\)

Exercises will help you prepare for the material covered in the next section. Evaluate each exponential expression in $$ \frac{x^{30}}{x^{-10}} $$

Explain how to simplify \(\sqrt{10} \cdot \sqrt{5}\)

Evaluate each expression without using a calculator. $$125^{\frac{2}{3}}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.