Exponents

Simplifying Square Roots and Rationalizing Denominators

Simplifying Square Roots (Review)

Let's review the steps involved in simplifying square roots:

1. Factor the number inside the square root sign.
2. If a factor appears twice, cross out both and write the factor one time to the left of the square root sign. If the factor appears three times, cross out two of the factors and write the factor outside the sign, and leave the third factor inside the sign.
   
   Note: If a factor appears 4 , 6 , 8 , etc. times, this counts as 2 , 3 , and 4 pairs, respectively.
   
3. Multiply the numbers outside the sign. Multiply the numbers left inside the sign.
4. Check: The outside number squared times the inside number should equal the original number inside the square root.

To simplify the square root of a fraction, simplify the numerator and simplify the denominator.

Example 1: Simplify

1. =
2. = 2×2
3. 2×2 = 4
4. Check: 4²(3) = 48

Thus, = 4 .

Example 2: Simplify .

First, reduce the fraction to lowest terms: =
=

Numerator:

1. =
2. = 2
3. 2 = 2
4. Check: 2²(3) = 12

Denominator:

1. =
2. = 5
3. 5 = 5
4. Check: 5²(7) = 175

Thus, = .

Rationalizing the Denominator

In addition to simplifying the numerator and the denominator in a fraction, it is mathematical convention to rationalize the denominator--that is, to write the fraction as an equivalent expression with no roots in the denominator.

To rationalize a denominator, multiply the fraction by a "clever" form of 1 --that is, by a fraction whose numerator and denominator are both equal to the square root in the denominator. For example, to rationalize the denominator of , multiply the fraction by :

× = = = .

Thus, = .

Often, the fraction can be reduced:

Rationalize the denominator of :

× = = = = 3 .

Thus, = 3 .

Occasionally, the resulting square root can be further simplified:

Rationalize the denominator of :

× = = = = = .

Thus, = .