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Chapter 6: Problem 13

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

- Identify and Write Down the Problem

The given problem is to multiply the fractions \(\frac{t-4}{8} \cdot \frac{4 t^{2}}{t-4}\). Write this problem down as it is.
02

- Factor and Simplify

Notice that \(t-4\) appears in both the numerator and the denominator. You can simplify by canceling out \(t-4\). This gives us \(\frac{4t^{2}}{8}\).
03

- Simplify the Remaining Fraction

Now, simplify \(\frac{4t^{2}}{8}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4. This results in \(\frac{t^{2}}{2}\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This usually involves dividing the numerator and the denominator by their common factors. For example, if you have \(\frac{4}{8}\), you can divide both 4 and 8 by their greatest common factor, which is 4. This will give you \(\frac{1}{2}\).
In the context of the given problem, the fraction \(\frac{4 t^{2}}{8}\) is simplified by dividing both the numerator and the denominator by 4, which is their greatest common divisor (GCD).
Knowing how to simplify fractions is essential because it makes calculations easier and results more readable.
Factoring
Factoring is the process of breaking down an expression into its simplest parts. For example, if you have \((a^2 - b^2)\), you can factor this into \((a + b)(a - b)\).
In our given problem, notice that \((t-4)\) appears in the numerator of one fraction and the denominator of another. This commonality allows us to cancel out \((t-4)\) from both fractions.
This step simplifies the problem and makes it easier to solve.
Therefore, factoring helps in identifying and eliminating common terms, making complex problems more manageable.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. For instance, for the numbers 12 and 16, the GCD is 4.
In the problem we are tackling, the GCD of the numerator and the denominator of \(\frac{4t^{2}}{8}\) is 4. By dividing both parts by their GCD, we simplify the fraction to its lowest terms.
Understanding how to find the GCD can greatly simplify your work with fractions and make your answers easier to interpret.
To find the GCD, you can list out the factors of each number and choose the highest one that appears in both lists.

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Most popular questions from this chapter

If we write \(\frac{3}{4}\) as an equivalent fraction with denominator \(28,\) by what number are we actually multiplying the fraction?

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{12}{x+2}+2}{\frac{18}{x+2}-2} $$

Let \(P\), \(Q\), and \(R\) be rational expressions defined as follows. $$P=\frac{6}{x+3}, \quad Q=\frac{5}{x+1}, \quad R=\frac{4 x}{x^{2}+4 x+3}$$ Perform the operations and express \(P+Q-R\) in lowest terms.

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ r+\frac{r}{4-\frac{2}{6+2}} $$
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