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Chapter 11: Problem 61

Solve each equation. Check your solutions. $$ r^{2 / 3}+r^{1 / 3}-12=0 $$

Short Answer

Expert verified
The solutions are \( r = -64 \) and \( r = 27 \).

Step by step solution

01

Substitute Variable

Let鈥檚 make a substitution to simplify the equation. Define a new variable,\( u = r^{1/3} \). Therefore, \( u^2 = r^{2/3} \). This transforms the equation into a quadratic form:\[ u^2 + u - 12 = 0 \].
02

Solve the Quadratic Equation

Solve the quadratic equation \( u^2 + u - 12 = 0 \). To do this, factor the quadratic equation:\( u^2 + u - 12 = (u + 4)(u - 3) = 0 \). Therefore, the solutions for \( u \) are:\[ u + 4 = 0 \Rightarrow u = -4 \] and \[ u - 3 = 0 \Rightarrow u = 3 \].
03

Substitute Back

Now, substitute \( u = r^{1/3} \) back into the solutions: For \( u = -4 \), we have \( r^{1/3} = -4 \) and for \( u = 3 \), we have \( r^{1/3} = 3 \).
04

Solve for r

Solve for \( r \) by raising both sides of the equations to the power of 3: For \( r^{1/3} = -4 \), we get \( r = (-4)^3 = -64 \). For \( r^{1/3} = 3 \), we get \( r = 3^3 = 27 \).
05

Check the Solutions

Substitute both values of \( r \) back into the original equation to verify they are correct: For \( r = -64 \), \( (-64)^{2/3} + (-64)^{1/3} - 12 = 16 - 4 - 12 = 0 \). For \( r = 27 \), \( 27^{2/3} + 27^{1/3} - 12 = 9 + 3 - 12 = 0 \). Both solutions satisfy the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method helps simplify complex equations. In this problem, the equation involves radical expressions. To make it easier to solve, we introduce a new variable.

For example, we let \( u = r^{1/3} \). This transforms the radical expression into a simpler quadratic form. At this point, it looks like \[ u^2 + u - 12 = 0 \].

This approach allows us to solve the equation using methods we are more familiar with, like factoring or using the quadratic formula. Afterwards, we can substitute back the values of \( u \) to find the original variable \( r \).
quadratic equations
Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). In our problem, after substitution, we have a quadratic equation \( u^2 + u - 12 = 0 \).

There are several methods to solve quadratic equations, including:
  • Factoring
  • Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • Completing the square

In this step, we chose factoring. We factored the quadratic equation into \( (u + 4)(u - 3) = 0 \), giving us the solutions \( u = -4 \) and \( u = 3 \).

Once the quadratic is solved, these solutions of \( u \) can be used for the next steps.
checking solutions
Checking solutions ensures that our answers are correct. This is especially important for radical equations, as we might have extraneous solutions.

To check, substitute the found values back into the original equation. For our problem:
  • Substitute \( r = -64 \): \( (-64)^{2/3} + (-64)^{1/3} - 12 \) should equal 0
  • Substitute \( r = 27 \): \( 27^{2/3} + 27^{1/3} - 12 \) should equal 0

By checking, we found that both \( r = -64 \) and \( r = 27 \) satisfy the original equation, confirming our solutions.

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