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Chapter 6: Problem 17

Add or subtract as indicated. $$\frac{9}{4 x^{3}}-\frac{17}{4 x^{3}}$$

Short Answer

Expert verified
\(\frac{-2}{x^3}\)

Step by step solution

01

Identify the common denominators

Both fractions have the same denominator, which is \(4x^3\).
02

Subtract the numerators

Since the denominators are the same, subtract the numerators directly: \(9 - 17 = -8\).
03

Combine the result

Place the result of the numerator subtraction over the common denominator: \(\frac{-8}{4x^3}\).
04

Simplify the fraction

Simplify \(\frac{-8}{4x^3}\): \(\frac{-8}{4x^3} = \frac{-2}{x^3}\).

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Key Concepts

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

Common Denominators
When subtracting fractions, the first step is to make sure that the fractions have the same denominator. Denominators are the bottom numbers of the fractions. For example, in the given exercise, both fractions \( \frac{9}{4x^3} \) and \( \frac{17}{4x^3} \) have the same denominator, which is \( 4x^3 \).
When denominators are the same, you can directly move to subtracting the numerators. However, if the denominators were different, you鈥檇 need to find a common denominator first. This usually involves finding the least common multiple of the denominators and then adjusting the fractions accordingly.
Numerator Subtraction
Once you have determined that the fractions have a common denominator, you can move on to subtracting the numerators. The numerator is the top part of the fraction.
In our given example, the numerators of the fractions are 9 and 17. To subtract these, simply subtract 17 from 9, which gives us -8: \( 9 - 17 = -8 \).
This computation is straightforward, but be mindful of the sign of the result, especially when the first numerator is smaller than the second.
You can write the result over the common denominator, which will give us \( \frac{-8}{4x^3} \).


Now that you鈥檝e subtracted the numerators, we can look at simplifying the fraction.
Fraction Simplification
Fraction simplification involves reducing the fraction to its simplest form. In our example, the fraction after subtraction is \( \frac{-8}{4x^3} \). To simplify it, divide both the numerator and the denominator by their greatest common divisor (GCD).
Here, the GCD of -8 and 4 is 4.
So, you divide both the numerator and the denominator by 4: \( \frac{-8}{4x^3} = \frac{-8 梅 4}{4x^3 梅 4} \).
This results in \( \frac{-2}{x^3} \).
That鈥檚 our simplified fraction! Simplifying fractions helps to make them easier to understand and work with in further calculations.

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Most popular questions from this chapter

Write each rational expression in lowest terms. $$ \frac{(x+4)(x-3)}{(x+5)(x+4)} $$

Write each rational expression in lowest terms. $$ \frac{a^{2}-b^{2}}{a^{2}+b^{2}} $$

Write each rational expression in lowest terms. $$ \frac{4 x(x+3)}{8 x^{2}(x-3)} $$

Consider the expressions \(\frac{x-2}{4}\) and \(\frac{2 x}{4} .\) Only one of these expressions is not in lowest terms. Which one is it and why? Write it in lowest terms.

Multiply or divide as indicated. $$ \frac{x^{2}-25}{x^{2}+x-20} \cdot \frac{x^{2}+7 x+12}{x^{2}-2 x-15} $$
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