/*! 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 51 Simplify the expression. Assume ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 51

Simplify the expression. Assume the letters denote any real numbers. $$\sqrt[3]{\sqrt{64 x^{6}}}$$

Short Answer

Expert verified
The simplified expression is \( 2x \).

Step by step solution

01

Simplify the Inner Radical

First, we start by simplifying the inner square root: \( \sqrt{64 x^6} \). We know that \( 64 \) is a perfect square because it equals \( 8^2 \). Also, \( x^6 \) is a perfect square because it equals \( (x^3)^2 \). Hence, \( \sqrt{64x^6} = \sqrt{(8x^3)^2} = 8x^3 \).
02

Apply the Cube Root

Now we apply the cube root to \( 8x^3 \). Since both \( 8 \) and \( x^3 \) are perfect cubes, we can express them as \( 8 = 2^3 \) and \( x^3 = (x)^3 \). Thus, \( \sqrt[3]{8x^3} = \sqrt[3]{2^3} \cdot \sqrt[3]{x^3} = 2x \).
03

Write the Final Simplified Expression

After applying the cube root, the expression simplifies to \( 2x \). This is the simplest form of the original expression, \( \sqrt[3]{\sqrt{64 x^6}} = 2x \).

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 of a number is one of three equal factors that, when multiplied together, give the original number.For a number \( a \), its cube root is denoted as \( \sqrt[3]{a} \).Understanding cube roots is important when simplifying expressions because:
  • It allows us to break down complex expressions into simpler parts.
  • A cube root of a perfect cube results in an integer, making calculations straightforward.
  • Simplifying cube roots involves looking for perfect cubes inside the radical.
For example, \( \sqrt[3]{8} = 2 \) because \( 2^3 = 8 \).This is helpful in algebraic expressions like \( \sqrt[3]{8x^3} \), where both 8 and \( x^3 \) are perfect cubes.The expression then simplifies to \( 2x \).
Make sure to identify perfect cubes to simplify expressions efficiently.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number.It's denoted by the radical symbol \( \sqrt{...} \), with the number inside called the radicand.Key aspects of square roots include:
  • Finding a square root is essentially doing the reverse of squaring a number.
  • If a number is a perfect square, its square root will be an integer.
  • Square roots are crucial in various fields, including geometry and algebra.
Consider the number 64, a perfect square since \( \sqrt{64} = 8 \).This plays a role in simplifying expressions like \( \sqrt{64x^6} \) because you recognize 64 as \( 8^2 \).
By identifying perfect squares within the radicand, you simplify square roots, aiding in algebraic manipulations.
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer.Recognizing perfect squares is handy for simplifying square roots and other mathematical operations.Important notes about perfect squares:
  • Numbers like 4, 9, 16, and 64 are perfect squares.
  • For example, \( 64 = 8^2 \), making it a perfect square.
  • Identifying perfect squares helps to simplify radicals in equations.
In the example \( \sqrt{64x^6} \), recognizing 64 as a perfect square allows a straightforward simplification.Similarly, \( x^6 \) is identified as a perfect square because \( (x^3)^2 = x^6 \).
Using perfect squares makes handling expressions under square roots much simpler.
Perfect Cubes
A perfect cube is a number that can be expressed as the cube of an integer.This concept is pivotal when simplifying cube roots within algebraic expressions.Key characteristics of perfect cubes include:
  • Numbers like 1, 8, 27, and 64 are perfect cubes.
  • This is because these numbers are obtained by raising integers like 1, 2, 3, and 4 to the power of 3.
  • Understanding perfect cubes simplifies the evaluation of cube roots.
In the expression \( \sqrt[3]{8x^3} \), knowing that 8 is \( 2^3 \) simplifies the cube root calculation.The same logic applies to \( x^3 \), which is already in the form of a perfect cube, \( (x)^3 \).
Spotting perfect cubes can significantly ease the simplification of algebraic expressions involving cube roots.

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

Use a Special Factoring Formula to factor the expression. $$27 x^{3}+y^{3}$$

(a) Factor the expressions completely: \(A^{4}-B^{4}\) and \(A^{6}-B^{6}\). (b) Verify that \(18,335=12^{4}-7^{4}\) and that \(2,868,335=12^{6}-7^{6}\). (c) Use the results of parts (a) and (b) to factor the integers \(18,335\) and \(2,868,335 .\) Then show that in both of these factorizations, all the factors are prime numbers.

Factor the trinomial. $$x^{2}-6 x+5$$

Factor the expression completely. (This type of expression arises in calculus when using the "product rule.") $$3(2 x-1)^{2}(2)(x+3)^{1 / 2}+(2 x-1)^{3}\left(\frac{1}{2}\right)(x+3)^{-1 / 2}$$

Make up several pairs of polynomials, then calculate the sum and product of each pair. Based on your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials?
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.