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Observe the following examples; then try to speculate on the method.

3 4 = ? 20 .

The original denominator 4 was multiplied by 5 to yield 20. What arithmetic process will yield 5 using 4 and 20?

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9 10 = ? 10 y .

The original denominator 10 was multiplied by y to yield 10 y .

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6 x y 2 a 3 b = ? 16 a 5 b 3 .

The original denominator 2 a 3 b was multiplied by 8 a 2 b 2 to yield 16 a 5 b 3 .

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5 a x ( a + 1 ) 2 = ? 4 ( a + 1 ) 2 ( a 2 ) .

The original denominator ( a + 1 ) 2 was multiplied by 4 ( a 2 ) to yield 4 ( a + 1 ) 2 ( a 2 ) .

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To determine the quantity that the original denominator was multiplied by to yield the new denominator, we ask, "What did I multiply the original denominator by to get the new denominator?" We find this factor by dividing the original denominator into the new denominator.

It is precisely this quantity that we multiply the numerator by to build the rational expression.

Sample set a

Determine N in each of the following problems.

8 3 = N 15 . The original denominator is 3 and the new  denominato is 15 . Divide the original  denominator into the new denominator and  multiply the numerator 8 by this result . 15 ÷ 3 = 5 Then, 8 5 = 40. So, 8 3 = 40 15 and N = 40. Check by reducing 40 15 .

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2 x 5 b 2 y = N 20 b 5 y 4 . The original denominator is  5 b 2 y  and the new  denominator is  20 b 5 y 4 . Divide the original  denominator into the new denominator and  multiply the numerator  2 x  by this result . 20 b 5 y 4 5 b 2 y = 4 b 3 y 3 So, 2 x · 4 b 3 y 3 = 8 b 3 x y 3 . Thus, 2 x 5 b 2 y = 8 b 3 x y 3 20 b 5 y 4 and N = 8 b 3 x y 3 .

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6 a a + 2 = N ( a + 2 ) ( a 7 ) . The new denominator divided by the original denominator is  ( a + 2 ) ( a 7 ) a + 2 = a 7 Multiply  6 a by a 7. 6 a ( a 7 ) = 6 a 2 + 42 a 6 a a + 2 = 6 a 2 + 42 a ( a + 2 ) ( a 7 )  and  N = 6 a 2 + 42 a .

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3 ( a 1 ) a 4 = N a 2 16 . The new denominator divided by the original denominator is a 2 16 a 4    = ( a + 4 ) ( a 4 ) a 4 = a + 4 Multiply  3 ( a 1 ) by a + 4. 3 ( a 1 ) ( a + 4 ) = 3 ( a 2 + 3 a 4 ) = 3 a 2 9 a + 12 3 ( a 1 ) a 4 = 3 a 2 9 a + 12 a 2 16 and N = 3 a 2 9 a + 12

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7 x = N x 2 y 3 . Write  7 x  as  7 x 1 . 7 x 1 = N x 2 y 3 Now we can see clearly that the original denominator  1 was multiplied by  x 2 y 3 .  We need to multiply the  numerator  7 x  by  x 2 y 3 . 7 x = 7 x · x 2 y 3 x 2 y 3 7 x = 7 x 3 y 3 x 2 y 3  and  N = 7 x 3 y 3 .

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5 x x + 3 = 5 x 2 20 x N . The same process works in this case .  Divide the original  numerator  5 x  into the new numerator  5 x 2 20 x . 5 x 2 20 x 5 x = 5 x ( x 4 ) 5 x = x 4 Multiply the denominator by  x 4. ( x + 3 ) ( x 4 ) 5 x x + 3 = 5 x 2 20 ( x + 3 ) ( x 4 ) and N = 5 x 2 20.

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4 x 3 x = N x 3 . The two denominators have nearly the same terms ;  each has  the opposite sign .  Factor  1  from the original denominator . 3 x = 1 ( 3 + x ) = ( x 3 ) 4 x 3 x = 4 x ( x 3 ) = 4 x x 3  and  N = 4 x .

It is important to note that we factored 1 from the original denominator. We did not multiply it by 1 . Had we multiplied only the denominator by 1 we would have had to multiply the numerator by 1 also.

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Practice set a

Determine N .

9 a 5 b = N 35 b 2 x 3

N = 63 a b x 3

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2 y y 1 = N y 2 1

N = 2 y 2 2 y

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a + 7 a 5 = N a 2 3 a 10

N = a 2 + 9 a + 14

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4 a = N 6 a 3 ( a 1 )

N = 24 a 4 ( a 1 )

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2 x = N 8 x 3 y 3 z 5

N = 16 x 4 y 3 z 5

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6 a b b + 3 = N b 2 + 6 b + 9

N = 6 a b 2 + 18 a b

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3 m m + 5 = 3 m 2 18 m N

N = m 2 m 30

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2 r 2 r 3 = 2 r 3 + 8 r 2 N

N = r 2 7 r + 12

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8 a b 2 a 4 = N 4 a

N = 8 a b 2

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The reason for building rational expressions

Building rational expressions

Normally, when we write a rational expression, we write it in reduced form. The reason for building rational expressions is to make addition and subtraction of rational expressions convenient (simpler).

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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