Problem 89 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{1}{2 x^{3}}-\frac{4}{3 x^{2}} $$

Step-by-Step Solution

Answer

\(\frac{3 - 8x}{6x^3}\)

Step 1: Identify the Least Common Denominator (LCD)

The denominators are \(2x^3\) and \(3x^2\). To add or subtract fractions, we need a common denominator. The least common denominator can be found by taking the highest power of each factor in the denominators. Here, we have:- The highest power of 2 is \(2\).- The highest power of 3 is \(3\).- The highest power of \(x\) is \(x^3\).Therefore, the LCD is \(6x^3\).

Step 2: Adjust the First Fraction

To bring the first fraction \(\frac{1}{2x^3}\) to a common denominator of \(6x^3\), we multiply both the numerator and the denominator by 3:\[\frac{1 \cdot 3}{2x^3 \cdot 3} = \frac{3}{6x^3}\].

Step 3: Adjust the Second Fraction

To bring the second fraction \(\frac{4}{3x^2}\) to a common denominator of \(6x^3\), we need to multiply both the numerator and the denominator by 2x:\[\frac{4 \cdot 2x}{3x^2 \cdot 2x} = \frac{8x}{6x^3}\].

Step 4: Subtract the Fractions

Now both fractions have the same denominator, so we can subtract them:\[\frac{3}{6x^3} - \frac{8x}{6x^3} = \frac{3 - 8x}{6x^3}\].

Step 5: Simplify (if needed)

The numerator \(3 - 8x\) and the denominator \(6x^3\) have no common factors other than 1, so the fraction is already in its simplest form.The final simplified answer is:\[\frac{3 - 8x}{6x^3}\].

Key Concepts

Understanding the Least Common Denominator

Whenever you encounter fractions, especially in operations like addition or subtraction, it's essential to find a common ground for the denominators—this is known as the least common denominator (LCD).
The LCD is the smallest number that each of the original denominators can divide into without leaving a remainder. It's like finding a mutual agreement between fractions so they can "speak the same language."

• When dealing with algebraic fractions, the LCD is found by identifying the highest power of each factor present in all the denominators. For instance, in the problem, the denominators are \(2x^3\) and \(3x^2\).
• For numbers, compare the coefficients: the highest power of each is 2 and 3. For variables, pick the one with the higher power: \(x^3\). By combining these, we find that the LCD is \(6x^3\).

Once you find the LCD, you can adjust all fractions involved to have this denominator. It paves the way for operations like subtraction to occur smoothly.

The Art of Fraction Subtraction

Subtracting fractions isn't as straightforward as whole numbers, primarily due to differing denominators. After finding the LCD, like we did above, the next step is to make sure that each fraction shares this common denominator.

• First, adjust each fraction so that their denominators match the LCD. For \( \frac{1}{2x^3} \), multiplying both the numerator and the denominator by 3 gives us \( \frac{3}{6x^3} \). Similarly, multiply \( \frac{4}{3x^2} \) by \(2x\) to get \( \frac{8x}{6x^3} \).
• Once the fractions are compatible with the LCD, you can subtract the numerators directly while keeping the denominator constant. This transforms into a simple arithmetic operation: subtract \(8x\) from 3.

The outcome is a single fraction with the overall expression \( \frac{3 - 8x}{6x^3} \). It's always the numerators that change; the denominator remains the same throughout. This method keeps things tidy and clear.

Simplifying Fractions: Final Touch

The last step in dealing with fractions is simplifying, if possible. Simplification involves reducing the fraction such that it's expressed in its simplest form.

• This process means checking if the numerator and denominator have common factors. If they do, divide them appropriately to reduce the fraction.
• In our expression, \(\frac{3 - 8x}{6x^3}\), examine the numerator \(3 - 8x\) and the denominator \(6x^3\). There are no shared factors other than 1, which means this fraction can't be reduced further.

By ensuring that each fraction is expressed in its simplest form, not only do you follow best mathematical practices, but it also makes the expression cleaner and easier to interpret. Simplifying is like the polishing step to reveal the most elegant form of your answer.