/*! 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 61 Simplify each expression. \(\s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 61

Simplify each expression. \(\sqrt[3]{y^{3}}\)

Short Answer

Expert verified
y

Step by step solution

01

Understand the problem

The expression given is \(\root{3}{y^3}\). The goal is to simplify this expression.
02

Apply the properties of exponents

Recall that the cube root of a number raised to the third power is the number itself. In mathematical terms, \(\root{3}{x^3} = x\).
03

Simplify the expression

Using the property from Step 2, apply it to the given expression \(\root{3}{y^3}\). This simplifies to \(\root{3}{y^3} = y\).

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 root
The cube root, denoted as \(\root{3}{x}\) or \(\root{3}{x}\), is a concept that allows you to find a number which, when multiplied three times by itself, gives the original number. For example, the cube root of 27 is 3 because \(\root{3}{27} = 3\) and \((3 \times 3 \times 3 = 27)\).
In simpler terms, the cube root is the inverse operation of cubing a number.
This means that taking the cube root of a number reverses the process of raising it to the power of three.
properties of exponents
Exponents are mathematical notations that denote repeated multiplication of a base number. Properties of exponents simplify operations involving powers. Some key properties include:
  • Product of Powers: \(a^{m} \times a^{n} = a^{m+n} \)
  • Power of a Power: \((a^{m})^{n} = a^{mn}\)
  • Cube Root Property: For any non-negative number a, \(\root{3}{a^{3}} = a\)

Using these properties helps to simplify complex expressions. For example, \(\root{3}{y^{3}}\) directly simplifies to \(y\) due to the cube root property.
simplification
Simplification involves rewriting expressions in the simplest form. In this context, we used the cube root and the properties of exponents to find the simplest version of \(\root{3}{y^{3}}\). Here鈥檚 a breakdown:
  • Recognize the expression: \(\root{3}{y^{3}} \)
  • Apply the cube root property: According to the property, \(\root{3}{y^{3}} = y\).

Thus, simplifying means making the expression cleaner and easier to work with, ultimately making it as straightforward as \(y\). Always look out for power and root relationships in your 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

Perform the indicated operations and simplify. $$ \left(2 a^{2}+11 b^{3}\right)^{2} $$

Add or subtract as indicated and simplify. $$ \left(-8 p^{7}-4 p^{4}+2 p-5\right)+\left(2 p^{7}+6 p^{4}+p^{2}\right) $$

Suppose that \(x\) represents the larger of two consecutive odd integers. a. Write a polynomial that represents the smaller integer. b. Write a polynomial that represents the sum of the two integers. Then simplify. c. Write a polynomial that represents the product of the two integers. Then simplify. d. Write a polynomial that represents the difference of the squares of the two integers. Then simplify.

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. a. Determine the total expenditure for private health insurance premiums for the year \(2010 .\) b. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2010 .\) c. Evaluate the polynomial \(I+P\) found in Exercise \(57(\) a) for \(x=10\).

Simplify each expression. $$ \left(x^{n}+3\right)\left(x^{n}-7\right) $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.