Rationalize Denominator
How to rationalize the denominator (monomial, binomial radical): 4 examples and their solutions.
Example√7y/√x
The denominator √x is a radical.
To rationalize the denominator √x,
multiply [√x/√x].
√7y⋅√x = √7xy
√x⋅√x = x
Multiply Radicals
So
√7xy/x
is the answer.
Example√0.2
0.2 = 2/10
2/10 = 1/5
√1/5
= √1/√5
= 1/√5
Divide Radicals
The denominator √5 is a radical.
To rationalize the denominator √5,
multiply [√5/√5].
1⋅√5 = √5
√5⋅√5 = 5
So
√5/5
is the answer.
Example1/[4 + √3]
The denominator 4 + √3
is a binomial radical.
To rationalize the denominator,
multiply, the conjugate of 4 + √3, 4 - √3
to both of the numerator and the denominator.
(The conjugate of [a + b] is [a - b].
The conjugate of [a - b] is [a + b].)
1⋅(4 - √3)
= 4 - √3
(4 + √3)(4 - √3)
= 42 - (√3)2
= 16 - 3
Product of a Sum and a Difference: (a + b)(a - b)
16 - 3 = 13
So
[4 - √3]/13
is the answer.
Example√2/[5 - √6]
The denominator 5 - √6
is a binomial radical.
To rationalize the denominator,
multiply, the conjugate of 5 - √6, 5 + √6
to both of the numerator and the denominator.
√2(5 + √6)
= 5√2 + √2⋅√6
Common Monomial Factor
(5 - √6)(5 + √6)
= 52 - (√6)2
= 25 - 6
√6 = √2⋅√3
25 - 6 = 19
√2⋅√2 = 2
So
[5√2 + 2√3]/19
is the answer.