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Chapter 4: Problem 49

Reduce each of the following rational expressions to lowest terms. $$\frac{4 s-3 t}{8 s-9 t}$$

Short Answer

Expert verified
The expression \(\frac{4s - 3t}{8s - 9t}\) is already in its lowest terms.

Step by step solution

01

Identify the greatest common factor (GCF)

Look for the greatest common factor of both the numerator (4s - 3t) and the denominator (8s - 9t). In this case, the terms do not have common factors other than 1.
02

Simplify the fraction

Since there are no common factors, the fraction \(\frac{4s - 3t}{8s - 9t}\) is already in its lowest terms.

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Key Concepts

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

greatest common factor
The greatest common factor (GCF) is the largest factor that two or more numbers share. Finding the GCF helps in simplifying expressions and fractions by identifying common terms that can be divided out.
For example, the GCF of 8 and 12 is 4 since 4 is the largest number that divides both 8 and 12 without leaving a remainder.
In algebra, we look for the GCF in the coefficients and any similar variables in the terms.
In the given problem, the expression is \( \frac{4s-3t}{8s-9t} \).
We try to identify common factors in both the numerator (4s - 3t) and the denominator (8s - 9t).
In this case, there are no shared factors other than 1. Therefore, the GCF is 1.
simplifying fractions
Simplifying fractions means reducing them to their simplest form so that the numerator and denominator have no common factors other than 1.
The process involves dividing both the top (numerator) and the bottom (denominator) of the fraction by their greatest common factor.
Here鈥檚 a quick example:
  • Consider the fraction \( \frac{10}{15} \).
  • The GCF of 10 and 15 is 5.
  • Divide both the numerator and the denominator by 5: \( \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \).
This fraction is now simplified.
In the exercise, since the numerator \(4s - 3t\) and the denominator \(8s - 9t\) have no common factors other than 1, the fraction is already in its simplest form.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and operations (such as addition and multiplication).
For example, in the expression \(4s - 3t\), 4 and -3 are coefficients; s and t are variables.
When reducing rational expressions, look at the algebraic expressions in the numerator and denominator.
In our problem, the numerator is \(4s - 3t\) and the denominator is \(8s - 9t\).
The goal is to find any common factors they share.
If there are no common factors, the fraction is already in its lowest terms, as we found out for \( \frac{4s-3t}{8s-9t} \).
Understanding how to handle algebraic expressions is key to simplifying and solving problems in algebra.

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. John and Susan leave their homes at 8: 00 A.M., going toward each other along the same route, which is 9 miles long. John walks at 4 mph while Susan jogs at 8 mph. At what time will they meet?

Make up a verbal problem involving money that would give rise to the following equation (be sure to indicate what the variable would represent): $$25 x+10(x+3)=275$$

Use a calculator to do the computation. Round off to three decimal places where necessary. $$\frac{1}{8} \cdot\left(\frac{2}{5} \div \frac{1}{2}\right)$$

Solve the equations and inequalities. $$z+\frac{z+5}{3}-\frac{z-2}{6}=\frac{z+4}{4}+1$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{4 y}{3 x^{2}}-\frac{3}{2 x}+\frac{y}{x^{2}}$$
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