/*! 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 124 What difference is there in simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 124

What difference is there in simplifying \(\sqrt[3]{(-5)^{3}}\) and \(\sqrt[4]{(-5)^{4}} ?\)

Short Answer

Expert verified
Simplification results in \(\sqrt[3]{(-5)^{3}} = -5\) and \(\sqrt[4]{(-5)^{4}} = 5\).

Step by step solution

01

Simplify the first root

To simplify \(\sqrt[3]{(-5)^{3}}\), we have to raise -5 to the power of 3, and then calculate the cube root which is the same as the base number. Therefore \(\sqrt[3]{(-5)^{3}} = -5.\)
02

Simplify the second Root

To simplify \(\sqrt[4]{(-5)^{4}}\), we have to raise -5 to the power of 4, and then calculate the fourth root. Here, taking an even root will result in a positive number, therefore \(\sqrt[4]{(-5)^{4}} = 5.\)
03

Comparison

While \(\sqrt[3]{(-5)^{3}}\) results in -5, \(\sqrt[4]{(-5)^{4}}\) equals 5. This shows the different results of simplifying odd and even roots of similar expressions.

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.

Cube Roots
Cube roots involve finding a number which, when multiplied by itself three times, gives the original number. It's like asking, "What number gives us this number when raised to the power of three?"
For instance, to simplify \( \sqrt[3]{(-5)^3} \):
  • First, calculate \((-5)^3\), which equals \(-125\).
  • The cube root of \(-125\) is \(-5\) since \((-5) \times (-5) \times (-5) = -125\).
Here, the cube root of a negative number results in another negative number, which is key to understanding cube roots. This is because the cube of a negative number remains negative.
Fourth Roots
Fourth roots are about finding a number which, when raised to the power of four, equals the original number. So, when you work with \( \sqrt[4]{(-5)^4} \):
  • Compute \((-5)^4\), which results in \(625\).
  • The fourth root of \(625\) is \(5\) because \(5 \times 5 \times 5 \times 5 = 625\).
Taking even roots, such as the fourth root, of a negative number raised to an even power always leads to a positive number. This happens because negative signs cancel out when multiplied an even number of times.
Negative Bases
Working with negative bases in roots and powers can be tricky, but breaking it down helps.
  • When you raise a negative base to an odd power (like 3), the result remains negative. For example, \((-5)^3 = -125\).
  • In contrast, raising a negative base to an even power (like 4), results in a positive number. For instance, \((-5)^4 = 625\).
Understanding this is fundamental to working with expressions that have negative bases, especially when considering their roots.
Simplifying Expressions
Simplifying expressions with roots involves breaking down each component to make them easier to handle. Key steps include:
  • Evaluate the power of the base first. Whether it's \((-5)^3\) or \((-5)^4\), compute this part of the expression first.
  • Next, find the appropriate root (cube or fourth). This means calculating \(\sqrt[3]{-125}\) or \(\sqrt[4]{625}\).
This step-by-step breakdown makes simplifying expressions methodical. Recognize the difference between cube roots yielding negatives and fourth roots yielding positives when working with negative bases.

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

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.

Will help you prepare for the material covered in the next section. If 6 is substituted for \(x\) in the equation $$ 2(x-3)-17=13-3(x+2) $$ is the resulting statement true or false?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. The toll to a bridge is \(\$ 3.00 .\) A three-month pass costs \(\$ 7.50\) and reduces the toll to \(\$ 0.50 .\) A six-month pass costs \(\$ 30\) and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

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.