/*! 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 35 Solve. $$y^{1 / 3}-y^{1 / 6}-6... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 35

Solve. $$y^{1 / 3}-y^{1 / 6}-6=0$$

Short Answer

Expert verified
The solution is y = 729

Step by step solution

01

Substitute variables

Introduce a new variable to simplify the equation. Let us set: \[ x = y^{1/6} \] Thus: \[ y^{1/3} = (y^{1/6})^2 = x^2 \]. The original equation becomes: \[ x^2 - x - 6 = 0 \]
02

Solve the quadratic equation

The equation \( x^2 - x - 6 = 0 \) can be factored into: \[ (x - 3)(x + 2) = 0 \] Thus, the solutions for \( x \) are: \[ x = 3 \] and \[ x = -2 \]
03

Substitute back the variable

Recall that \( x = y^{1/6} \). Therefore for \( x = 3 \), we have: \[ y^{1/6} = 3 \] which means \[ y = 3^6 = 729 \] And for \( x = -2 \), it does not make sense for \(y\) to be a real number because \( y^{1/6} \) cannot be negative.

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.

Substitution Method
The substitution method is a useful technique where you introduce a new variable to simplify an equation. In this exercise, we substitute:
Let: x = y^{1/6} This simplifies y^{1/3} as: y^{1/3} = (y^{1/6})^2 = x^2 This transforms our original complex expression into a simpler quadratic equation: x^2 - x - 6 = 0 The substitution method helps break down complicated expressions into easier-to-manage parts. This method is particularly helpful when dealing with higher-order polynomials or nested functions. By substituting part of the equation with a new variable, you turn a possibly daunting problem into one that uses familiar, simpler techniques.
Quadratic Equations
Quadratic equations are polynomials of the form: ax^2 + bx + c = 0 In the given problem, our simplified equation is: x^2 - x - 6 = 0 Quadratic equations can be solved using factoring, the quadratic formula, or by completing the square. Here, we use factoring. We factorize x^2 - x - 6 into: (x - 3)(x + 2) = 0 Next, we set each factor to zero: x - 3 = 0 x + 2 = 0 This gives us the solutions: x = 3 x = -2 Factoring is a straightforward method when the quadratic easily breaks down into binomials. Always remember:
  • Combine like terms
  • Look for common factors
  • Set each factor to zero to solve for the variable
Exponent Rules
Exponent rules are fundamental when dealing with powers and roots of numbers. Some key rules to remember are:
  • a^m * a^n = a^{m+n}
  • (a^m)^n = a^{m*n}
  • a^0 = 1 (for a 鈮� 0)
  • a^{-m} = 1/a^m
In this context, we deal specifically with roots and fractional exponents. Recall: y^{1/3} = (y^{1/6})^2 This makes y^{1/3} an exponent of a power. After solving the quadratic equation x^2 - x - 6 = 0 and finding x = 3 or x = -2, we substitute back: x = y^{1/6}. When x = 3: y^{1/6} = 3 To isolate y, raise both sides to the power of 6: y = 3^6 = 729. Understanding how to manipulate exponents is foundational for simplifying equations and solving polynomial 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

Factor completely. $$ 3 y^{3}-300 y $$

Wyatt is tied to one end of a 40-m elasticized (bungee) cord. The other end of the cord is secured to a winch at the middle of a bridge. If Wyatt jumps off the bridge, for how long will he fall before the cord begins to stretch? (Use \(4.9 t^{2}=s\).)

Solve each formula for the indicated letter. Assume that all variables represent positive numbers. \(A=6 s^{2},\) for \(s\) (Surface area of a cube)

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$f(x)=x^{4}-4 x^{3}-x^{2}+16 x-12$$

Solve. Canoeing. During the first part of a canoe trip, Ken covered \(60 \mathrm{km}\) at a certain speed. He then traveled \(24 \mathrm{km}\) at a speed that was \(4 \mathrm{km} / \mathrm{h}\) slower. If the total time for the trip was 8 hr, what was the speed on each part of the trip?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.