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Equivalent Rational Expressions
• When we multiply both the numerator and the
denominator of a rational number by the same non-zero
value, we create an equivalent rational number.
• Example 1:
3
4

35
45

15
20
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3
4

35
45

15
20
Equivalent Rational Numbers
Note that when we multiply both numerator and
denominator by 5, we are actually multiplying
the fraction by 1.
The result is that the value of the fraction is not
changed.
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• When we multiply both the numerator and the
denominator of a rational expression by the same non-zero
expression, we create an equivalent rational expression.
• Example 2:
3 2 y
6y
3


5x 5 x  2 y 10 xy
Equivalent Rational Expressions
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• Our goal is often to create an equivalent rational
expression with a given denominator.
• Example 3:
Write the first rational expression as an equivalent
rational expression with the given denominator.
3

5 x 10 x3 y 5
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3

3 5
5 x 10 x y
Determine what you would multiply times the
denominator on the left to get the denominator on the
right.
3 5
5 x  ???  10 x y
5 2  10
3
2
x x  x
5
5
y  y
The required factor is
2
2x y
5
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Multiply both numerator and denominator by the
expression …
3  2x y
2 5
5x  2 x y
2
5
… to get the equivalent rational expression with the
required denominator.
2
5
6x y

3 5
10 x y
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• Example 4:
Write the first rational expression as an equivalent
rational expression with the given denominator.
5x

2
2
4 x  16 24  x  2  x  2 
Factor the denominator of the rational expression
on the left.


4 x  16  4 x  4  4  x  2 x  2
2
2
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5x

2
4  x  2  x  2  24  x  2  x  2 
Determine what you would multiply times the
denominator on the left to get the denominator on the
right.
We need
6
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5x

2
4  x  2  x  2  24  x  2  x  2 
Determine what you would multiply times the
denominator on the left to get the denominator on the
right.
We need
6  x  2
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Multiply both numerator and denominator by the
expression …
5x  6  x  2
4  x  2  x  2  6  x  2 
… to get the equivalent rational expression with the
required denominator.

30 x  x  2 
24  x  2  x  2 
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2
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