/*! 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 42 Simplify. See Examples 3 and 4 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 42

Simplify. See Examples 3 and 4 $$ \sqrt[3]{64 y^{9}} $$

Short Answer

Expert verified
The simplified form is \(4y^3\).

Step by step solution

01

Understand the expression

We need to simplify the expression \( \sqrt[3]{64 y^9} \). This is a cube root of a product of a constant (64) and a variable term \( y^9 \).
02

Simplify the constant term

Consider the cube root of 64. We need to find a number whose cube gives 64. We know that \( 4^3 = 4 \times 4 \times 4 = 64 \), so \( \sqrt[3]{64} = 4 \).
03

Simplify the variable term

Consider the cube root of \( y^9 \). We can rewrite \( y^9 \) as \( (y^3)^3 \), so \( \sqrt[3]{y^9} = \sqrt[3]{(y^3)^3} = y^3 \).
04

Combine simplified components

Combine the simplified components: the cube root of the constant is 4 and the cube root of the variable term is \( y^3 \). Thus, \( \sqrt[3]{64 y^9} = 4y^3 \).

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
A cube root is an important algebraic concept. It involves finding a number that, when multiplied by itself twice, gives the original number. For example, the cube root of 64 means looking for a number that, when cubed, equals 64. In this scenario, it's 4 because \( 4^3 = 4 \times 4 \times 4 = 64 \).
  • Cube root is represented as \( \sqrt[3]{x} \).
  • It鈥檚 the inverse operation of cubing (raising to the third power).
  • Finding cube roots helps in simplifying expressions involving cubic terms.
When dealing with cube roots in expressions, it's crucial to identify whether you're working with perfect cubes or not. Simplifying cube roots involves recognizing these perfect cubes and breaking them down, just like in the original exercise with 64.
Simplifying Expressions
Simplifying expressions is a critical skill in algebra that makes expressions easier to work with. It involves reducing them to their simplest form. In the exercise, the expression is \( \sqrt[3]{64 y^9} \). We simplify it by addressing each part separately.
  • First, consider any constant terms, like 64, and find their simplified form, which is 4 here.
  • Then, handle any variable terms systematically.
By reducing each part, expressions become less complex and more manageable. This can involve factoring, expanding, or using arithmetic operations. Always simplify piece by piece and then recombine all simplified sections to find the final reduced form.
Variable Terms
Variable terms in algebra often involve powers, like \( y^9 \) in our exercise. When simplifying, especially involving roots, it's essential to recognize patterns in exponents. Here, \( y^9 \) can be rewritten as \( (y^3)^3 \), which makes calculating its cube root straightforward.
  • Identify how exponents can be reorganized to simplify operations like roots.
  • Use the rule that \( \sqrt[3]{(x^3)} = x \) when dealing with cube roots.
  • Apply this to simplify variable terms effectively.
In this exercise, converting \( y^9 \) to \( (y^3)^3 \) allows its cube root to be simply \( y^3 \), demonstrating how radical and exponent rules can be used to manage and simplify variable terms efficiently.

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 each exponential expression. $$\frac{\left(2 a^{-1} b^{2}\right)^{3}}{\left(8 a^{2} b\right)^{-2}}$$

For Exercises 105 through \(108,\) do not use a calculator. \(\sqrt{160}\) is closest to \(\begin{array}{lll}{\text { a. } 10} & {\text { b. } 13} & {\text { c. } 20}\end{array}\) d. 40

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as \(8 \cdot 3,\) because 8 is a perfect cube. $$ 54 $$

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{a^{2}}}{\sqrt[6]{a}} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

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.