Show the similarities between dividing two fractions and dividing two rational expressions using examples of each; explain your steps
In both cases, invert the bottom one and multiply.
1/2
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3/5
= 1/2 * 5/3 = 5/6
a/b
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p/q
= a/b * q/p = (a q) / (b p)
Thanks Damon!
To understand the similarities between dividing two fractions and dividing two rational expressions, let's start by explaining how to divide each of them individually.
Dividing Two Fractions:
To divide two fractions, you need to multiply the first fraction by the reciprocal of the second fraction. Here's an example:
Example 1: Divide 2/3 by 4/5.
Solution: To divide 2/3 by 4/5, we multiply the first fraction by the reciprocal of the second fraction, which means flipping the second fraction. It becomes:
(2/3) ÷ (4/5) = (2/3) × (5/4)
Now, multiply across numerator and denominator:
= (2 × 5) / (3 × 4)
= 10/12
Simplifying this fraction gives us 5/6 as the final answer.
Dividing Two Rational Expressions:
Dividing two rational expressions follows the same concept as dividing fractions. The only difference is that instead of having numbers, we deal with algebraic expressions. Here's an example:
Example 2: Divide (x^2 + 3x - 2) / (x^2 - 1) by (2x + 1) / (x^2 - 4).
Solution: To divide these rational expressions, we multiply the first expression by the reciprocal of the second expression:
[(x^2 + 3x - 2) / (x^2 - 1)] ÷ [(2x + 1) / (x^2 - 4)]
= [(x^2 + 3x - 2) / (x^2 - 1)] × [(x^2 - 4) / (2x + 1)]
Now, we'll simplify and cancel out any common factors between the numerators and denominators. Factorize the quadratic expressions:
= [(x + 2)(x - 1) / (x + 1)(x - 1)] × [(x + 2)(x - 2) / (2x + 1)]
= [(x + 2)(x - 2)(x + 2)] / [(x + 1)(x - 1)(2x + 1)]
Since (x - 1) cancels out in the numerator and denominator, the final expression becomes:
= (x + 2)(x + 2) / (x + 1)(2x + 1)
Simplifying this rational expression is not possible since it cannot be further reduced.
Similarities:
The similarities between dividing two fractions and dividing two rational expressions can be summarized as follows:
1. Both divisions involve multiplying the first expression by the reciprocal of the second expression.
2. The process aims to simplify the fractions or rational expressions by canceling out common factors in the numerator and denominator.
Note that the examples provided demonstrate the general method of dividing fractions and rational expressions. In practice, the problems may vary in complexity, and additional steps may be required for simplification or finding restrictions on variables.