/*! 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 28 Simplify. $$ \sqrt[3]{-c^{6}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 28

Simplify. $$ \sqrt[3]{-c^{6}} $$

Short Answer

Expert verified
\((-c^{2})\)

Step by step solution

01

Understand the Cube Root

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. For cube roots, the symbol is \( \sqrt[3]{...} \).
02

Simplify the Expression Inside the Cube Root

The expression inside the cube root is \( -c^{6} \). We can rewrite this as \( (-1) \cdot (c^{6}) \).
03

Take the Cube Root of -1

The cube root of \( -1 \) is \( -1 \) because \( (-1) \cdot (-1) \cdot (-1) = -1 \). So, \( \sqrt[3]{-1} = -1 \).
04

Simplify the Cube Root of \\( c^{6} \\\\)

Taking the cube root of \( c^{6} \) gives us \( c^{6/3} = c^{2} \). This is because \( c^{6} = (c^{2})^{3} \).
05

Combine the Results

Combine the results from Steps 3 and 4: \( \sqrt[3]{-c^{6}} = (-1) \cdot c^{2} = -c^{2} \).

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
Understanding cube roots is essential to simplifying expressions like \( \sqrt[3]{-c^6} \). A cube root helps us find a number that, when multiplied by itself twice, gives us the original value. For example, if we have \( x = \sqrt[3]{y} \), then \( x \cdot x \cdot x = y \). Cube roots are represented using the symbol \( \sqrt[3]{...} \). Here, the '3' indicates that we are trying to find a cube root, as opposed to a square root which would be \( \sqrt{...} \).
  • Cubes involve multiplying a number three times by itself.
  • Cube roots help reverse this operation, revealing the original base number.
In our example, \( \sqrt[3]{-c^{6}} \), we applied cube roots to both \(-1\) and \(c^6\). The expression inside the cube root is decomposed and simplified individually, making it easier to work with complex numbers.
Exponents
Exponents are a way to express repeated multiplication of the same number. For instance, \(a^6\) means \(a \cdot a \cdot a \cdot a \cdot a \cdot a\). They are crucial in scientific notation and algebra for simplifying complex expressions. When dealing with cube roots like in our example \(\sqrt[3]{-c^6}\), it's important to understand how exponents interact with roots.
  • Using exponents enables us to express large numbers more easily.
  • They are tightly connected with roots, helping us simplify roots like cube roots and square roots.
In the expression \(c^6\), we used the relationships between exponents and cube roots. To solve \(\sqrt[3]{c^6}\), convert the exponents into fractional form: \(c^{6/3} = c^2\). Here, the cube root extracts a third root, effectively simplifying the expression.
Negative Numbers
Negative numbers are numbers less than zero. They are represented with a minus sign \(-\) in front. Understanding them is important in algebra and complex calculations. In the context of cube roots, negative numbers have interesting properties. Let's explore.
  • Negative numbers can be multiplied three times to give a negative result, unique to cube roots.
  • They often appear in real-life calculations, adding depth to graph plots and real processes.
In our solved exercise with \(\sqrt[3]{-1}\), this property of negative numbers helps us directly. The cube root of a negative number like \(-1\) is still \'-1\' because \((-1) \cdot (-1) \cdot (-1)\) matches \(-1\). This is a critical feature allowing us to work smoothly with negatives within cube root expressions.

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. $$ \sqrt[4]{(x-5)^{2}} $$

Evaluate each expression. \(\frac{1}{8}+\frac{5}{12}\)

Solve each inequality. $$ \sqrt{y-7}+5 \geq 10 $$

Solve each inequality. $$ 6-\sqrt{2 y+1}<3 $$

Suppose a culture has 100 bacteria to begin with and the number of bacteria doubles every 2 hours. Then the number \(N\) of bacteria after \(t\) hours is given by \(N=100 \cdot 2^{\frac{t}{2}} .\) How many bacteria will be present after 3\(\frac{1}{2}\) hours?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

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.