Divide.
$\frac{x^2-36}{12 x} \div \frac{6-x}{4 x y} = $ (Simplify your answer.)
Answer (1)
∙ Rewrite the division as multiplication by the reciprocal: 12 x x 2 − 36 ⋅ 6 − x 4 x y .
∙ Factor x 2 − 36 as ( x − 6 ) ( x + 6 ) and rewrite 6 − x as − ( x − 6 ) .
∙ Substitute the factored forms into the expression and cancel common factors. ∙ Simplify the expression to obtain the final answer: − 3 ( x + 6 ) y .
Explanation
Understanding the Problem We are asked to divide two rational expressions and simplify the result. The given expression is 12 x x 2 − 36 ÷ 4 x y 6 − x .
Rewriting the Division To divide by a fraction, we multiply by its reciprocal. Thus, we have: 12 x x 2 − 36 ÷ 4 x y 6 − x = 12 x x 2 − 36 ⋅ 6 − x 4 x y
Factoring Next, we factor the numerator x 2 − 36 as a difference of squares: x 2 − 36 = ( x − 6 ) ( x + 6 ) Also, we can rewrite 6 − x as − ( x − 6 ) .
Substituting Factors Substituting these factorizations into the expression, we get: 12 x ( x − 6 ) ( x + 6 ) ⋅ − ( x − 6 ) 4 x y
Canceling Common Factors Now, we cancel common factors. We can cancel the factor ( x − 6 ) from the numerator and denominator, and we can also cancel the factor x . This gives us: 12 ( x + 6 ) ⋅ − 1 4 y
Simplifying Constants We can simplify the constants by dividing 4 by 12, which gives us 3 1 . So we have: 3 ( x + 6 ) ⋅ − 1 y = − 3 ( x + 6 ) y
Final Answer Thus, the simplified expression is − 3 ( x + 6 ) y
Examples
Rational expressions are useful in many areas, such as physics, engineering, and economics. For example, in physics, they can be used to describe the relationship between voltage, current, and resistance in an electrical circuit. Suppose the total resistance in a parallel circuit is given by a rational expression. Simplifying such expressions helps in analyzing the circuit's behavior and designing it efficiently. Similarly, in economics, rational functions can model cost-benefit ratios, where simplifying them aids in making informed decisions about resource allocation and investment strategies.
