/*! 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 64 Write an equivalent expression w... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 64

Write an equivalent expression with negative exponents. $$ (-6)^{4} $$

Short Answer

Expert verified
\(\frac{1}{(-6)^{-4}}\).

Step by step solution

01

Understand Negative Exponents

Negative exponents represent the reciprocal of the positive exponent. For example, \((a^{-n}) = \frac{1}{a^n}\). To rewrite a positive exponent with a negative exponent, you can use this information.
02

Rewrite the Expression with Negative Exponents

Given \((-6)^4\), you can express this with a negative exponent by taking the reciprocal. The equivalent expression with a negative exponent is: \(\frac{1}{(-6)^-4}\).
03

Verify the Expression

To confirm the equivalence, recognize that \(\frac{1}{(-6)^-4}\) indeed simplifies back to \((-6)^4\).

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.

Reciprocal
When dealing with exponents, it鈥檚 essential to understand the concept of a reciprocal. The reciprocal of a number is simply one divided by that number. For instance, the reciprocal of 5 is \(\frac{1}{5}\). When we talk about negative exponents, we are actually using the concept of reciprocals.
The rule is: \(a^{-n} = \frac{1}{a^n}\). This means that any number raised to a negative exponent is equal to one divided by that number raised to the opposite positive exponent.
  • Example: \((2^{-3} = \frac{1}{2^3} = \frac{1}{8})\)

In our exercise, we need to express \((-6)^4\) using negative exponents. By the reciprocal rule, we get \(\frac{1}{(-6)^{-4}}\).
Equivalent Expressions
An equivalent expression is a way of writing the same value or quantity in a different form. In mathematics, equivalent expressions are crucial for simplifying problems and finding solutions.
When we rewrite \((-6)^4\) as \(\frac{1}{(-6)^{-4}}\), we are creating an equivalent expression. Even though they look different, both expressions represent the same value. To verify that these two expressions are equivalent, you can reverse the operation.
  • If \frac{1}{(-6)^{-4}}\ simplifies back to \((-6)^4\), then they are indeed equivalent.

Remember that the goal of using equivalent expressions is not to change the value but to provide a form that might be more useful in a given context.
Positive Exponents
When working with exponents, the starting point is usually positive exponents. These indicate how many times to multiply the base number by itself. For example, \(3^2\) means multiplying 3 by itself: \(3 \times 3 = 9\).
Positive exponents are easier to understand because they reflect straightforward multiplication. In our problem, \((-6)^4\) means multiplying -6 by itself four times, which results in 1296.
  • Step-by-step: \(-6 \times -6 = 36\), \(36 \times -6 = -216\), \(-216 \times -6 = 1296\).
By converting it to a negative exponent, we don't change the value but instead use a different form. This helps us understand the linkage between positive and negative exponents.
Recognizing these patterns is vital for solving more complex mathematical problems.

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 \(\left(a^{m}\right)^{n}\) $$ \left(a^{3}\right)^{2} $$

Use roster notation to write each set. The set of all natural numbers that are multiples of 5

To the student and the instructor: Writing exercises, denoted by \([\%,\) are meant to be answered using sentences. Because answers to many writing exercises will vary, solutions are not listed at the back of the book. Charlie insists that \(15-4+1 \div 2 \cdot 3\) is \(2 .\) What error is he making?

Use mathematical symbols to translate each phrase. Six less than half of a number

Divide. $$ \frac{-28}{-7} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.