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Chapter 3: Problem 75

Solve each equation. Find imaginary solutions when possible. $$(3 u-1)^{2 / 3}=2$$

Short Answer

Expert verified
The solution is \( u = \frac{2 \sqrt{2} + 1}{3} \). There are no imaginary solutions.

Step by step solution

01

Isolate the Power Expression

Given the equation \( (3u - 1)^{2/3} = 2 \), isolate \( (3u - 1) \) by raising both sides of the equation to the power of \( \frac{3}{2} \): \[ ( (3u - 1)^{2/3} )^{3/2} = 2^{3/2} \]\[ 3u - 1 = 2^{3/2} \]
02

Simplify the RHS

Calculate \ 2^{3/2} \. \[ 2^{3/2} = (\sqrt{2})^3 = 2 \sqrt{2} \]Now the equation is: \[ 3u - 1 = 2 \sqrt{2} \]
03

Solve for u

Add 1 to both sides of the equation: \[ 3u = 2 \sqrt{2} + 1 \]Next, divide both sides by 3: \[ u = \frac{2 \sqrt{2} + 1}{3} \]
04

Check for Imaginary Solutions

Determine if there are any imaginary solutions. Since all manipulations were valid and kept us in the real number domain, there are no imaginary solutions for this equation.

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Key Concepts

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

Imaginary Solutions
Imaginary solutions often arise when dealing with equations that include square roots of negative numbers. However, not all equations will have imaginary solutions. Some equations, like the one in this exercise, remain within the bounds of real numbers without encountering negative discriminants or complex roots. To identify these types of solutions, ensure every step maintains the equation within the real number domain.
Isolate the Variable
. This effectively removed the fractional exponent. Once the variable term (3u-1) was isolated, we further simplified the equation to solve for u. Breaking down complex steps into simpler tasks, like isolating the variable, makes it easier to manage and solve the equation systematically.
Simplify Expressions
By simplifying the right-hand side, we made it easier to perform further algebraic manipulations. Simplifying ensures clarity and helps prevent errors when solving complex equations.
Real and Complex Numbers
where a is the real part and b is the imaginary part. In certain equations, like quadratic equations with negative discriminants, the solutions may be complex numbers. However, the equation in the exercise remains strictly within the real number domain, as all operations only involved real numbers. It's important to understand when complex numbers are required, and how they apply within algebraic contexts.

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