Rationalizing a Denominator

Rationalizing a Denominator

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

Example 1

Simplify the given expression. Square root (x^3)/(8(y^2)).

Split the radical sign into two parts:
the numerator and the denominator.

Simplify the radicals
in the numerator and the denominator.

Simplifying a radical (part 2)

To rationalize √2 in the denominator,
multiply √2
to both of the numerator and the denominator.

Then √2⋅√2 = 2.
So the denominator is rationalized.

Example 2

Simplify the given expression. Square root 0.2.

To rationalize √5,
multiply √5
to both of the numerator and the denominator.

Example 3

Simplify the given expression. 1 / (4 + square root 3).

To rationalize 4 + √3,
multiply its conjugate, 4 - √3,
to both of the numerator and the denominator.

Then (4 + √3)(4 - √3) = 42 - (√3)2 = 13.

The denominator is rationalized.

Product of a sum and a difference

4 + √3 and 4 - √3
are the conjugates of each other.

Conjugates: a + b and a - b.

Example 4

Simplify the given expression. (square root 2) / (5 - square root 6).

To rationalize 5 - √6,
multiply its conjugate, 5 + √6,
to both of the numerator and the denominator.

Numerator: √2(5 + √6) = 5√2 + 2√3

Multiplying radicals (part 2)

Example 5

Simplify the given expression. (x - 4) / (square root x - 2).

To rationalize √x - 2,
multiply its conjugate, √x + 2,
to both of the numerator and the denominator.

Cancel (x - 4).

Simplifying rational expressions