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Simplify the complex rational expression by using the LCD. [Be sure to remove any common factors.]
((7)/(y)+(y)/(r))/((r)/(y)-(9)/(r))

Simplify the complex rational expression by using the LCD. [Be sure to remove any common factors.] $\newline$ $\frac{\frac{7}{y}+\frac{y}{r}}{\frac{r}{y}-\frac{9}{r}}$

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Multiplication with rational exponents

Full solution

Q. Simplify the complex rational expression by using the LCD. [Be sure to remove any common factors.]

\newline

\frac{\frac{7}{y}+\frac{y}{r}}{\frac{r}{y}-\frac{9}{r}}

Identify LCD: Identify the least common denominator (LCD) for the complex rational expression. $\newline$ The expression is $\left(\frac{7}{y} + \frac{y}{r}\right) / \left(\frac{r}{y} - \frac{9}{r}\right)$ . The denominators are $y$ , $r$ , $y$ , and $r$ . The LCD for these denominators is $y*r$ .
Multiply by LCD: Multiply both the numerator and the denominator of the complex fraction by the LCD to clear the denominators. $\newline$ We have: $\newline$ $\left(\frac{7}{y} + \frac{y}{r}\right) / \left(\frac{r}{y} - \frac{9}{r}\right) \times \frac{y\cdot r}{y\cdot r}$
Distribute LCD: Distribute the LCD to each term in the numerator and the denominator. $\newline$ $\frac{(\frac{7}{y}) \cdot (y\cdot r) + (\frac{y}{r}) \cdot (y\cdot r)}{(\frac{r}{y}) \cdot (y\cdot r) - (\frac{9}{r}) \cdot (y\cdot r)}$
Simplify terms: Simplify each term by canceling out the common factors. $\newline$ $[\frac{(7 \cdot r) + (y^2)}{(r^2) - (9 \cdot y)}]$
Combine terms: Combine the terms in the numerator and the denominator. $\newline$ The numerator is already simplified as $(7r + y^2)$ . The denominator is also simplified as $(r^2 - 9y)$ . There are no common factors to remove.
Write final expression: Write the final simplified expression. $\newline$ The simplified complex rational expression is $(7r + y^2) / (r^2 - 9y)$ .

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