Problem 63 - 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. $$\frac{\frac{2}{3}(x-2) x^{-1 / 3}-x^{2 / 3}}{(x-2)^{2}}$$

Step-by-Step Solution

Answer

The expression simplifies to zero at \(x = -4\).

Step 1: Simplify the Expression

We start by simplifying the expression so it doesn't have any negative exponents.The given expression is:\[\frac{\frac{2}{3}(x-2) x^{-1 / 3}-x^{2 / 3}}{(x-2)^{2}}\]First, we will rewrite the term \(x^{-1/3}\) as \(\frac{1}{x^{1/3}}\):\[\frac{\frac{2}{3}(x-2) \frac{1}{x^{1/3}} - x^{2/3}}{(x-2)^{2}}\]Now, multiply \(\frac{2}{3}(x-2)\) by \(\frac{1}{x^{1/3}}\):\[\frac{\frac{2}{3(x^{1/3})}(x-2) - x^{2/3}}{(x-2)^{2}}\]

Step 2: Simplify Further

Now, we simplify further by combining the terms in the numerator:To do this, find a common denominator, which is \(x^{1/3}\):\[\frac{\frac{2(x-2) - 3x}{3x^{1/3}}}{(x-2)^{2}}\]Combine the terms in the numerator:\[= \frac{\frac{2x - 4 - 3x}{3x^{1/3}}}{(x-2)^{2}}\]This simplifies to:\[= \frac{\frac{-x - 4}{3x^{1/3}}}{(x-2)^{2}}\]

Step 3: Set the Expression Equal to Zero

Now that we've simplified the expression (excluding negative exponents), we set it to zero to solve for \(x\):\[\frac{\frac{-x - 4}{3x^{1/3}}}{(x-2)^{2}} = 0\]For a fraction to be equal to zero, the numerator must be zero:\[\frac{-x - 4}{3x^{1/3}} = 0\]Simplify the condition to solve:\(-x - 4 = 0\).

Step 4: Solve for x

To solve \(-x - 4 = 0\), we add 4 to both sides of the equation:\(-x = 4\)Next, multiply both sides by -1 to solve for \(x\):\(x = -4\).Thus, the solution is \(x = -4\).

Key Concepts

Negative Exponents

Negative exponents can seem a bit tricky at first, but they're actually quite simple! Let's break it down. When you see an expression like \(x^{-n}\), you can rewrite this as \(\frac{1}{x^n}\). Essentially, a negative exponent tells you to take the reciprocal of the base raised to the positive of that exponent.
For example, \(x^{-1/3}\) becomes \(\frac{1}{x^{1/3}}\). This is a very useful technique because it allows us to work with fractions instead of negative powers, which makes simplifying expressions much easier.
Whenever you come across negative exponents in an equation, remember:

• Flip the base to the denominator (or numerator if it was originally in the denominator).
• Change the sign of the exponent to positive.

Once you have rewritten all negative exponents as positive, your expression becomes much simpler to solve or further manipulate!

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that will help you solve complex problems with ease. When simplifying an algebraic expression, your goal is to reduce it to its simplest form by combining like terms and eliminating unnecessary components.
The first step is to identify and rewrite any negative exponents, as discussed earlier. After handling the exponents, it’s time to look at the terms in the expression.
Here’s how you can simplify expressions:

• Identify common denominators to combine fractions.
• Combine like terms by adding or subtracting coefficients.
• Reduce fractions by dividing both the numerator and the denominator by their greatest common factor.

In the provided solution, after converting the negative exponent, we identified common denominators to combine the terms in the numerator before simplifying the fraction. This makes the expression more manageable and clear.

Solving Equations

Once an expression is simplified, it may be part of an equation that needs solving. Solving equations is all about finding the value of the unknown variable—usually \(x\).
To solve equations effectively:

• First, ensure the equation is in its simplest form by eliminating negative exponents and simplifying all terms.
• Set the equation such that all variable terms are one side and constants on the other.
• Solve for the variable using basic algebraic operations like addition, subtraction, multiplication, or division.

In our specific exercise, after simplifying the expression without negative exponents, we set the simplified equation equal to zero. It led to solving a simple linear equation \(-x - 4 = 0\).
By rearranging terms and performing arithmetic operations, we found \(x = -4\).
Always ensure to check your solution by plugging it back into the original equation to see if it holds true!