Rationalizing a Denominator

Rationalizing a Denominator

How to rationalize the radicals in a denominator: examples and their solutions.

Example 1: Simplify √5xy/x2

Simplify the given expression. Square root [5xy/x^2]

Reduce x2 in the numerator to x
and cancel x in the denominator.

Split the radical sign.

Divide radicals

The denominator is √x,
which is an irrational number.

So this cannot be the answer.

To rationalize the denominator,
multiply (√x)/(√x).

5y⋅√x = √5xy

Multiply radicals

x⋅√x = x

So (given) = √5xy/x.

The denominator is rationalized.
So this is the answer.

Example 2: Simplify √0.2

Simplify the given expression. Square root [0.2]

0.2 = 2/10

So √0.2 = √2/10.

Cancel 2
and reduce 10 to 5.

Then (given) = √1/5.

The numerator is 1.

So the radical sign only affects the denominator.

So (given) = 1/√5.

The denominator is √5,
which is an irrational number.

So, to rationalize the denominator,
multiply (√5)/(√5).

5⋅√5 = 5

So (given) = √5/5.

Conjugate

The conjugate of [a + b] is [a - b]. The conjugate of [a - b] is [a + b]. (a + b)(a - b) = a^2 - b^2 is used to rationalize the radical denominator.

The conjugate of [a + b] is [a - b].
The conjugate of [a - b] is [a + b].

So [a + b] and [a - b]
are the conjugates of each other.

Recall that (a + b)(a - b) = a2 - b2.

Product of a sum and a difference (a + b)(a - b)

The product of a binomial and its conjugate
makes a2 - b2.

By using this property,
you can rationalize a binomial denominator.

Example 3: Simplify 1/(4 + √3)

Simplify the given expression. 1/(4 + Square root [3])

The denominator is [4 + √3],
which is an irrational binomial.

The conjugate of [4 + √3] is [4 - √3].

So, to rationalize the denominator,
multiply (4 - √3)/(4 - √3).

(4 + √3)(4 - √3) = 42 - (√3)2

Product of a sum and a difference (a + b)(a - b)

42 = 16

(√3)2 = 3

Square root - Square of a square root

16 - 3 = 13

So (given) = (4 - √3)/13.

Example 4: Simplify √2/(5 - √6)

Simplify the given expression. Square root [2]/(5 - Square root [6])

The denominator is [5 - √6],
which is an irrational binomial.

The conjugate of [5 - √6] is [5 + √6].

So, to rationalize the denominator,
multiply (5 + √6)/(5 + √6).

2⋅(5 + √6) = 5√2 + √2⋅√6

(5 - √6)(5 + √6) = 52 - (√6)2

2⋅√6 = √2⋅√2⋅√3

52 = 25

(√6)2 = 6

2⋅√2⋅√3 = 2√3

25 - 6 = 19

So (given) = (5√2 + 2√3)/19.

Example 5: Simplify (x - 4)/(√x - 2)

Simplify the given expression. (x - 4)/(Square root [x] - 2)

The denominator is [√x - 2],
which is an irrational binomial.

The conjugate of [√x - 2] is [√x + 2].

So, to rationalize the denominator,
multiply (√x + 2)/(√x + 2).

Leave the numerator.
You might use (x - 4).

(√x - 2)(√x + 2) = (√x)2 - 22.

(√x)2 = x

22 = 4

Cancel (x - 4).

Then (given) = √x + 2.

Simplifying rational expressions