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Chapter 6: Problem 70

Multiply or divide as indicated. $$ \frac{(6 x+5)(x-3)}{(x-1)(x-3)} \div \frac{(2 x+7)(x+9)}{(x-1)(x+9)} $$

Short Answer

Expert verified
\[\frac{6x + 5}{2x + 7}\]

Step by step solution

01

- Write the expression

We start with the given expression: \[\frac{(6x+5)(x-3)}{(x-1)(x-3)} \bigg/ \frac{(2x+7)(x+9)}{(x-1)(x+9)}\]
02

- Simplify the division

To simplify a division of fractions, we multiply by the reciprocal of the second fraction. This means flipping the second fraction: \[\frac{(6x+5)(x-3)}{(x-1)(x-3)} \times \frac{(x-1)(x+9)}{(2x+7)(x+9)}\]
03

- Simplify the expression

Next, we cancel out the common factors in the numerator and the denominator: \[\frac{(6x+5) \bcancel{(x-3)}}{\bcancel{(x-1)} \bcancel{(x-3)}} \times \frac{\bcancel{(x-1)} \bcancel{(x+9)}}{(2x+7) \bcancel{(x+9)}}\] This reduces the expression to: \[\frac{(6x+5)}{(2x+7)}\]
04

- Final simplified expression

The simplified expression from the given multiplication and division operations is: \[\frac{6x + 5}{2x + 7}\]

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Key Concepts

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

simplifying rational expressions
Simplifying rational expressions involves reducing them to their most basic form. A rational expression is a fraction where both the numerator and the denominator are polynomials. To simplify, follow these steps:
  • **Factor both the numerator and the denominator**: Break them down into their prime factors.
  • **Cancel out common factors**: Any polynomial that appears in both the numerator and the denominator can be canceled.
For example, in the expression \(\frac{(6x+5)(x-3)}{(x-1)(x-3)}\), the factor \((x-3)\) appears in both the numerator and the denominator, so it can be canceled out. That leaves us with the simplified form \(\frac{6x+5}{x-1}\). Remember, always check for any restrictions on the variable to avoid division by zero.
multiplying fractions
When multiplying fractions, the process is straightforward: multiply the numerators together and the denominators together. The steps are:
  • **Write down both fractions**: Clearly write the formula you need to apply.
  • **Multiply the numerators**: Multiply the numbers or expressions in the top parts of both fractions.
  • **Multiply the denominators**: Do the same for the bottom parts.
  • **Simplify**: Reduce the resulting fraction if possible.
In the provided example, after flipping the second fraction and multiplying, you get: \(\frac{(6x+5)(x-3)}{(x-1)(x-3)} \times \frac{(x-1)(x+9)}{(2x+7)(x+9)}\). Multiply the terms in the numerators and denominators, then simplify to \(\frac{(6x+5)}{(2x+7)}\).
dividing fractions
Dividing fractions is slightly different from multiplying. To divide one fraction by another, you multiply the first fraction by the reciprocal of the second. Here's how:
  • **Take the reciprocal**: Flip the second fraction (the fraction you are dividing by).
  • **Multiply**: Proceed with multiplication as described above.
In the practice problem, we start with the division: \(\frac{(6x+5)(x-3)}{(x-1)(x-3)} \bigg/ \frac{(2x+7)(x+9)}{(x-1)(x+9)}\). Taking the reciprocal of the second fraction, we get: \(\frac{(6x+5)(x-3)}{(x-1)(x-3)} \times \frac{(x-1)(x+9)}{(2x+7)(x+9)}\). Multiply and simplify to the final result, \(\frac{6x+5}{2x+7}\).

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Most popular questions from this chapter

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

Write each rational expression in lowest terms. $$ \frac{r^{3}-s^{3}}{r-s} $$

Which two rational expressions are equivalent to \(-1 ?\) A. \(\frac{2 x+3}{2 x-3}\) B. \(\frac{2 x-3}{3-2 x}\) C. \(\frac{2 x+3}{3+2 x}\) D. \(\frac{2 x+3}{-2 x-3}\)

Write each rational expression in lowest terms. $$ \frac{t^{2}-9}{3 t+9} $$

Add or subtract as indicated. $$\frac{r+s}{3 r^{2}+2 r s-s^{2}}-\frac{s-r}{6 r^{2}-5 r s+s^{2}}$$
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