How to Divide Square Roots with Coefficients?

Square roots are type of Radicals with power over the root as 2. Dividing Real Numbers in general would result in either fraction of the form P /Q (rational) or can result in an Integer value. But the same is not possible with the Irrational Numbers every time as their division may result into another irrational number. Moreover if the irrational number is formed in the denominator, then Rationalization has to be used to make the denominator real valued. Let us see how to divide Square roots with coefficients with help of some examples:

Example 1: Divide 44 √14 by 54 √7?

Solution: Here two radicals 44 √14 and 54 √7 have 44 and 54 as coefficients so they will be divided separately irrespective of the irrationals which are to be divided independently. So,

44 √14 /54 √7 = (44 /54) * (√14 /√7) = (22 /27) * √ (14 /7) = (22 √7 /27),

Here, rationalization is not required as there are no radicals in the denominator.

Example 2: Divide 20 √5 by 4 √2?

Solution: Here also two radicals 20 √5 and 4 √2 have 20 and 4 as coefficients but their bases are different this time. So, by dividing the coefficients and irrational numbers separately we get:

20 √5 /4 √2 = (20 /4) * (√5 /√2) = (5) * √ (5 /2) = (5 √5 /√2),

Here, we see that the denominator is still irrational and therefore there is a need to do rationalize the denominator. This can be done as follows: Dividing and multiplying the fraction by (√2) we get:

(5 √5 /√2) * (√2) / (√2) = (5 √ (5 *2) /√ (2 * 2)) = 5 (√10) /2,

Thus, we get a real denominator as 2.