Rationalizing the Denominator of a Fraction
with a Monomial Term that Contains a Square Root

Each of following factions has a monomial expression in the denominator that contains a square
root or a square root multiplied by a number.

It is common to require that the denominator of a fraction not contain any radicals. The process of
eliminating radicals from the denominator of a fraction is called rationalizing the denominator.

Reducing Factions with Monomial terms like sqrt(A)/sqrt(B)

Many fractions with square roots in both the numerator and denominator can be simplified by reducing
the numbers under the square roots. Fractions with only monomial terms that are each under a square
root like sqrt(A)/sqrt(B) may be reduced if A and B have a common factor. If the numbers under the square
roots have a common factor then the numbers under the square roots can reduce each other.

Example 1 Simplify sqrt (12)/sqrt(3)	Example 2 Simplify sqrt(50)/sqrt(8)
sqrt(12)/sqrt(3) the 12 and 3 are both factors under a radicalsign so they can reduce each other	sqrt(50)/sqrt(8) the 50 and 8 are both factors under a radical sign so they can reduce each other

Reducing Factions with Monomial terms like

Fractions with only monomial terms like may be be reduced. If A and B have a common
factor then the numbers under the square root can reduce each other. If C and D have a common
factor then the numbers outside the square root can reduce each other.

Warning: A number under a square root and a number outside a square root CAN NOT reduce
each other

Example 3 Simplify	Example 4 Simplify
the 12 and 27 are both factors under a radical sign so they can reduce each other the 10 and 4 are both factors outside a radical sign so they can reduce each other	the 27 and 75 are both factors under a radical sign so they can reduce each other the 6 and 9 are both factors outside a radical sign so they can reduce each other

Rationalizing the Denominator of a Fraction with
a Monomial Term that contains a Square Root

Many fractions with square roots can not be simplified by reducing the numbers under the square
roots as in the examples above.

It is common to require that the denominator not contain any radicals. In fractions where the numbers
under the square square roots cannot be reduced to eliminate the square root in the denominator we
must find another process that will eliminate the radical from the denominator. The process of
eliminating the radical from the denominator of a fraction is called rationalizing the denominator.

Multiplying the fraction sqrt(A)/sqrt(B) by sqrt(B)/sqrt(B) will eliminate the radical from the denominator

(7)/sqrt(3) multiply the top and bottom by sqrt(3)	(5)/sqrt(6) multiply the top and bottom by sqrt(6)
Example 5 Simplify (7)/sqrt(3)	Example 6 Simplify (5)/sqrt(6)
(7)/sqrt(3) multiply the top and bottom by sqrt(3)	(5)/sqrt(6) multiply the top and bottom by sqrt(6)
the sqrt(3) times itself is 3	the sqrt(6) times itself is 6
Note: The 3 under the radical sign and the 3 outside the radical cannot reduce each other	Note: The 6 under the radical sign and the 6 outside the radical cannot reduce each other
Example 7 Simplify (6)/sqrt(10)	Example 8 Simplify (25)/sqrt(15)
(6)/sqrt(10) multiply the top and bottom by sqrt(10)	(25)/sqrt(15) multiply the top and bottom sqrt(15)
the sqrt(10) times itself is 10	the sqrt(15) times itself is 15
the 6 and 10 are both factors outside a radical sign so they can reduce each other	the 25 and 15 are both factors outside a radical sign so they can reduce each other
Example 9 Simplify	Example 10 Simplify
multiply the top and bottom by	multiply the top and bottom by
Example 13 Simplify	Example 14 Simplify
multiply the top and bottom by sqrt(6)	the 10 and 6 are both factors under a radical sign so they can reduce each other
	multiply the top and bottom by sqrt(3)
reduce4/6