Multiplication and Division of  Rational Expressions

15 Multiplication and Division of Rational Expressions

$\displaystyle{ \Big(\frac{P}{Q}\Big)^{-1} = \frac{Q}{P} \qquad \qquad \frac{P}{Q} . \frac{S}{T} = \frac{P.S}{Q.T} }$

$\displaystyle{ \frac{\frac{P}{Q}}{\frac{S}{T}} = \frac{P}{Q} . \Big(\frac{S}{T}} \Big)^{-1} = \frac{P}{Q} . \frac{T}{S} = \frac{P.T}{Q.S}$

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Exercise 1

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We have

$\displaystyle{\frac{ x- 4}{x^2-16}.\frac{ x+3}{x^2+16}. \frac{ x+4 }{6 x +18 } }$

$= \displaystyle{\frac{ x- 4}{ (x-4) (x+4) }.\frac{ x+3}{x^2+16}. \frac{ x+4 }{6 (x +3) } }$

$= \displaystyle{ \frac{ 1}{6(x^2+ 16) } . }$

Exercise 2

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We have

$\displaystyle{ \frac{\displaystyle{\frac{x -10}{ x^3-9x^2 }} }{\displaystyle{\frac{ x^2 -14x +40}{ x^2 -12x + 27 } }} }$

$=\displaystyle{\frac{x -10}{ x^3-9x^2 }. \frac{ x^2 -12x + 27 }{ x^2 -14x +40} }$

$= \displaystyle{\frac{x -10}{ x^2(x-9) }. \frac{(x-9)(x-3)}{ (x-10)(x-4) } }$

$= \displaystyle{\frac{x-3}{ x^2(x-4) } }$ .

Exercise 3

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We have

$\displaystyle{ \frac{\displaystyle{\frac{x^2 + 12 x +27}{ x^2 + 16 x + 60 }} }{\displaystyle{\frac{ x^2 +18x +81}{ x^2 +19x + 90 } }} }$

$=\displaystyle{\frac{x^2 + 12 x +27}{ x^2 + 16 x + 60 }. \frac{ x^2 +19x + 90 }{x^2 +18x +81} }$

$= \displaystyle{\frac{(x +9)(x+3)}{ (x+10)(x+6) }. \frac{(x+9)(x+10)}{ (x+9)(x+9) } }$

$= \displaystyle{\frac{x+3}{ x+6 } }$ .

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