/*! 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 12 In Exercises 3鈥�12, solve the e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 12

In Exercises 3鈥�12, solve the equation. Check your solution. $$ \sqrt[4]{4 x}-13=-15 $$

Short Answer

Expert verified
Attempting to solve the equation yields \(x = 4\), but this solution does not satisfy the original equation. Therefore, the equation has no solution in the real numbers.

Step by step solution

01

Isolate the fourth root term

Consider the equation \(\sqrt[4]{4x}-13=-15\). Isolate the fourth root term by adding 13 to both sides: \(\sqrt[4]{4x} = -15 + 13 = -2 \)
02

Resolve the fourth root

Eliminate the fourth root by raising both sides of the equation to the power of 4: \( (\sqrt[4]{4x})^4 = (-2)^4\), which simplifies to \(4x = 16\)
03

Solve for x

Finally, solve for \(x\) by dividing by 4 on both sides: \(x = 4\)
04

Check the solution

Substitute \(x = 4\) back into the original equation to verify the solution. \(\sqrt[4]{4*4} - 13 = \sqrt[4]{16} - 13 = 2 - 13 = -11\) which is not equal to -15. Therefore, the solution is invalid.

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.

Fourth Root
When working with equations involving the fourth root, it's essential to understand what the fourth root represents. The fourth root of a number is a value that, when multiplied by itself four times, gives the original number.
For example, the fourth root of 16 is 2 because \( 2^4 = 16 \). Fourth roots, like other roots, can be tricky because they involve radicals.
  • The notation \( \sqrt[4]{x} \) refers to the fourth root of \( x \).
  • A fourth root can have both a positive and a negative solution, e.g., \( \sqrt[4]{16} = \pm 2 \).
Understanding these basic concepts of fourth roots is crucial for solving equations that contain them and will help in resolving the equation in the correct manner.
Checking Solutions
Checking the solution of an equation is an important step in ensuring that the value found satisfies the original equation. Sometimes, due to the operations performed during solving, such as taking roots or powers, extraneous solutions can occur.
For example, when solving \( \sqrt[4]{4x} - 13 = -15 \), after finding \( x = 4 \), it's essential to substitute back into the original equation to verify the result.
  • Substitute the value you found back into the original equation.
  • Simplify to see if the equation holds true.
  • If it doesn't, then the solution might be extraneous, meaning it doesn't satisfy the original equation, even though the arithmetic was correct.
In this exercise, finding \( x = 4 \) didn't satisfy the original equation, indicating an extraneous solution after the operation.
Isolating Terms
Isolating terms in an equation is a fundamental step in solving it. The goal is to rearrange the equation in such a way that you have the variable of interest alone on one side of the equation. This often involves moving terms around using addition, subtraction, multiplication, or division.
Consider the equation \( \sqrt[4]{4x} - 13 = -15 \). Here, isolating the fourth root term is the first critical step.
  • Add or subtract terms on both sides to isolate the term with the variable.
  • In this example, adding 13 to both sides simplifies the equation to \( \sqrt[4]{4x} = -2 \).
  • Always perform inverse operations to achieve the variable isolation efficiently.
By understanding how to properly isolate terms, you can systematically tackle different components of an equation gradually, leading to the desired solution.

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

\(|x+8|=|2 x+2|\)

\(f(x)=2 \sqrt{x}, g(x)=f(x+3)\)

\(|2 x-6|=|x|\)

The surface area \(A\) (in square meters) of a person with a mass of 60 kilograms can be approximated by \(A=0.2195 h^{0.3964}\), where \(h\) is the height (in centimeters) of the person. a. Find the inverse function. Then estimate the height of a 60-kilogram person who has a body surface area of \(1.6\) square meters. b. Verify that function \(A\) and the inverse model in part (a) are inverse functions.

\(f(x)=\sqrt{x}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Probability and 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.