Problem 66 - A Graphical Approach to College Algebra

Question

Complete the following. (a) Simplify the given expression so that it does not have negative exponents. (b) Set the expression from part (a) equal to 0 and solve the resulting equation. $$\frac{\frac{2}{3}(3 x+2) x^{-1 / 3}-3 x^{2 / 3}}{(3 x+2)^{2}}$$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{-3x + 4}{3(3x+2)^2} \). Solution: \( x = \frac{4}{3} \).

Step 1: Simplify the numerator

The numerator of the expression is \( \frac{2}{3}(3 x+2) x^{-1 / 3} - 3 x^{2 / 3} \). To simplify, start by distributing \( \frac{2}{3} \) to \( (3 x + 2) \), yielding: \[ \left(2 + \frac{4}{3x^{1/3}}\right) - 3x^{2/3}. \] This turns the original numerator to \( \frac{2(3x + 2)}{3x^{1/3}} - 3x^{2/3} \).

Step 2: Eliminate negative exponents

To remove the negative exponent, \( x^{-1/3} \) is rewritten as \( \frac{1}{x^{1/3}} \). Multiplying all terms by \( x^{1/3} \) to eliminate the fraction gives: \( \frac{2(3x + 2)}{3x^{1/3} } \, \cdot \, x^{1/3} - 3x^{2/3} \, \cdot \, x^{1/3} \). This simplifies to: \[ 2(3x + 2)/3 - 3x. \]

Step 3: Simplify the expression further

Multiply out to get: \[ \frac{2(3x + 2)}{3} - 3x = \frac{6x + 4}{3} - 3x. \] Simplify by finding a common denominator: \[ \frac{6x + 4 - 9x}{3}. \] This reduces to: \[ \frac{-3x + 4}{3}. \]

Step 4: Solve the equation

Set the simplified expression equal to zero: \[ \frac{-3x + 4}{3(3x+2)^2} = 0. \] Simplify to find \( -3x + 4 = 0 \). Solve for \( x \) by moving terms around: \( -3x = -4 \), leading to \( x = \frac{4}{3} \).

Step 5: Verify the solution satisfies all conditions

In the context of the original expression's domain, note any restrictions where the denominator cannot be zero. However, the obtained solution \( x = \frac{4}{3} \) does satisfy the requirement as it does not make the denominator zero.

Key Concepts

Negative Exponents

Negative exponents can seem confusing at first, but they are actually quite simple to understand once you break them down. A negative exponent means that the base of the power is on the "wrong" side of a division line. You should think of negative exponents as reciprocals.
For example, an expression like \( x^{-1/3} \) can be rewritten in a more familiar, positive exponent form as \( \frac{1}{x^{1/3}} \). Notice how the base \( x \) shifts from the numerator to the denominator when the exponent changes from negative to positive.
You can eliminate negative exponents across an entire expression by multiplying with the base raised to the positive version of the same exponent. This way, you ensure that all your computations use positive exponents, which makes further calculations much easier. Remember to do this step by step for each term to ensure accuracy in your simplification.

Solving Equations

Solving equations is an essential process in algebra that allows us to find the value of unknown variables. In the context of simplifying expressions to solve equations, it's crucial to first simplify your equation as much as possible by combining like terms and getting rid of fractions or anything complicating the equation.
In our example, after simplifying the expression to \( \frac{-3x + 4}{3} \), we set it equal to zero since we seek the values of \( x \) which satisfy this condition. This simplifies our task to finding when the numerator is zero, because a fraction is zero when its numerator is zero, not the denominator. With \( -3x + 4 = 0 \), you solve for \( x \) by isolating it.

• Move terms around to get \( -3x = -4 \)
• Divide both sides by \(-3\) to find \( x = \frac{4}{3} \)

This yields the solution \( x = \frac{4}{3} \), which must be checked to ensure it doesn't violate any domain restrictions like making the denominator zero.

Algebraic Fractions

Working with algebraic fractions is similar to working with numerical fractions, but involves variables. You simplify algebraic fractions by finding common denominators to combine terms and reduce the fraction to its simplest form. The goal is to reduce the complexity of the fraction, making it easier to solve or further manipulate.
In the given expression \( \frac{-3x + 4}{3(3x+2)^2} \), notice the denominator \( 3(3x+2)^2 \) involves a polynomial expression. Before solving or simplifying further, always ensure the denominator doesn't equal zero, as this would make the fraction undefined.
For this exercise, after simplifying the numerator and ensuring it's not affected by the denominator becoming zero for any value of \( x \), you can effectively solve it. Always exercise care with algebraic fractions to maintain balance between numerator and denominator and not inadvertently break mathematical rules like division by zero.