Dividing radical

Question

How to divide radicals with a radical in the denominator.

Answer 1

Example 1: To simplify √2 / √5, multiply the numerator and denominator by √5 to get √10 / 5.

Example 2: To simplify √2 / (3 + √5), multiply the numerator and denominator by 3 - √5, the conjugate of

the denominator to get (3√2 - √10) / 4.

Answer 2

The convention of standard math is that you do not want a square root in the denominator. However, why is this the case? One of the many reasons we rationalize, or remove the square root, from the denominator is so because it makes the division much easier. Before calculators were invented, mathematicians needed ways to perform math quickly and easily. They found that dividing √2 by 2 was much easier than dividing 1 by √2 (if you doubt me, you should try. It is very hard). So, we have continued in their wise practices. Now, regarding the given question.

In order to rationalize √2 / √5 we must multiply √5 on the top and botome like this

√2 x √5

√5 √5

We can perform this multiplication because we are essentially multiplying √2 / √5 by 1, which does not change the value of the equation

We are now left with

√10

5

So, now you know why and how to rationalize fractions :)

Answer 3

Answer 4

When dividing radicals by another radical, you're going to need to apply the Quotient Rule to conduct the operation. The Quotient Rule is as follows:

√a / √b = √(a/b)

where a ≥ 0, b > 0

The rule states that "The square root of a quotient is equal to the quotient of the square roots of the numerator and denominator" or in simpler terms, dividing the square root of a by the square root of b is the same as taking the square root of a divided by b. If the radicals in the numerator and/or denominator are being multiplied by a number in front of them, then divide the coefficients. Here are a couple of examples that contain coefficients that are 1 or greater than 1:

Example 1

√24 / √8 = √(24/8) = √3

Example 2

25√16 / 5√2 = (25/5) x (√16 / √2) = (25/5) x [√(16/2)] = 5 x √(8) = 5 x √(4 x 2) = 5 x 2√2 = 10√2

Sometimes, as shown in example two, you may be able to simplify the radical by identifying the largest factor of the number within the radical that you can take the square root of and leave only that which you gives you an irrational number as an answer.

Something else to keep in mind when handling radicals is that it's not ideal to leave radicals in the denominator in any of your final answers. In order to remove it from the denominator, yu simply need to multiply the numerator and denominator by the radical in your denominator. Example 3 demonstrates this.

Example 3

4√5 / 7√3 = (4√5 / 7√3) x (√3 / √3) = (4√5x3) / (7√3x3) = 4√15 / 7√9 = 4√15 / (7 x 3) = 4√15 / 21