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Chapter 6: Problem 11

Multiply. Write each answer in lowest terms. $$ \frac{t-4}{8} \cdot \frac{4 t^{2}}{t-4} $$

Short Answer

Expert verified
\frac{t^2}{2}

Step by step solution

01

Factor the expressions

Before multiplying the fractions, look for common factors that can cancel out. Both the numerator and the denominator in the expressions can be simplified.
02

Cancel common factors

Notice that \( t-4 \) appears in both the numerator and the denominator. Cancel out \( t-4 \) from both fractions. This simplifies the expression to \[ \frac{1}{8} \cdot \frac{4t^{2}}{1} \]
03

Multiply the simplified fractions

Multiply the remaining numerators and denominators: \[ \frac{1}{8} \cdot \frac{4t^{2}}{1} = \frac{4t^2}{8} \]
04

Simplify the resulting fraction

Now simplify the fraction \[ \frac{4t^2}{8} = \frac{t^2}{2} \]. The numerator can be divided by 4 to reduce the fraction to its lowest terms.

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Key Concepts

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

Factoring Algebraic Expressions
Factoring involves writing an expression as a product of its factors. When multiplying algebraic fractions, factoring can help simplify the process. For example, in the exercise \(\frac{t-4}{8} \cdot \frac{4t^{2}}{t-4}\), notice that the numerator and denominator have a common term \((t-4)\). By factoring out \((t-4)\), we make it easier to cancel terms later.

Factoring steps:
  • Identify common factors in the numerator and the denominator.
  • Rewrite the expression by factoring these common terms.
Simplifying Fractions
Simplifying fractions means reducing them to their lowest terms. After factoring the expressions in the exercise, it's easier to see which terms can be canceled out. In fractions, simplifying involves:

  • Dividing both the numerator and the denominator by their greatest common factor (GCF).
  • Reducing the fraction so that the numerator and the denominator have no other common factors besides 1.
For instance, after canceling \(t-4\) from both the numerator and denominator in our exercise, we end up with \(\frac{1}{8} \cdot \frac{4t^{2}}{1}\).
Canceling Common Factors
Canceling common factors is a crucial step in simplifying algebraic fractions. This involves removing identical factors from the numerator and denominator. In the exercise, we had \(\frac{t-4}{8} \cdot \frac{4t^{2}}{t-4}\). The factor \((t-4)\) appears in both fractions, so we cancel it out.

Steps to cancel common factors efficiently:
  • Factor both the numerator and the denominator to their simplest forms.
  • Identify and remove identical factors from the numerator and the denominator.
  • Multiply the resulting simplified fractions as shown: \( \frac{1}{8} \cdot \frac{4t^{2}}{1} = \frac{4t^2}{8} \).
  • Further simplify the result if possible. Here, \(\frac{4t^2}{8} = \frac{t^2}{2}\).

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Most popular questions from this chapter

If the rational expression \(\frac{5 x^{2} y^{3}}{2 p q}\) represents the area of a rectangle and \(\frac{2 x y}{p}\) represents the length, what rational expression represents the width?

\(\begin{array}{llll}{\text { What is the least common denominator for } \frac{9}{20} \text { and } \frac{1}{2} ?} & {} & {} \\ {\text { A. } 40} & {\text { B. } 2} & {\text { C. } 20} & {\text { D. none of these }}\end{array}\)

Identify the following as an expression or an equation. Then simplify the expression or solve the equation. \(\frac{3}{5} x-\frac{7}{10} x\)

Find the greatest common factor of each group of terms. $$ 24 m, 18 m^{2}, 6 $$

Find the LCD for the fractions in each list. $$ \frac{12}{m-3}, \frac{-4}{3-m} $$
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