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Chapter 7: Problem 65

Perform the indicated operation or operations. $$\left(\frac{y-2}{y^{2}-9 y+18} \cdot \frac{y^{2}-4 y-12}{y+2}\right) \div \frac{y^{2}-4}{y^{2}+5 y+6}$$

Short Answer

Expert verified
The simplified expression is \(\frac{(y-6)(y^2 + 5y + 6)}{(y-2)(y-3)}\).

Step by step solution

01

Factor the Expressions

Begin by factoring the expressions. Using the quadratic factoring approach, we get: \[ \left(\frac{(y-6)(y+3)}{(y-6)(y-3)} \cdot \frac{(y-6)(y+2)}{(y+2)}\right) \div \frac{(y-2)(y+2)}{(y+3)(y+2)} \] After factoring, cancel out the common factors.
02

Perform the Multiplication and Division Operations

Dividing by a fraction is the same as multiplying by its reciprocal. Rearrange and rewrite the expression to get: \[ \left(\frac{(y-6)(y+3)}{(y-3)}\right) \cdot \left(\frac{(y+3)(y+2)}{(y-2)}\right) \]
03

Simplify the Result

Perform the multiplication and simplify the expression to get the final result. \[ \frac{(y-6)(y^2 + 5y + 6)}{(y-2)(y-3)} \]

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Most popular questions from this chapter

add or subtract as indicated. Simplify the result, if possible. $$\frac{6 y^{2}+y}{2 y^{2}-9 y+9}-\frac{2 y+9}{2 y^{2}-9 y+9}-\frac{4 y-3}{2 y^{2}-9 y+9}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{2}{y+5}+\frac{3}{4 y}$$

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x-2}{x^{2}-25}-\frac{x-2}{25-x^{2}}$$

Describe how to identify the corresponding sides in similar triangles.
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