Home » Math Theory » Algebra » Division of Radicals

## Introduction

We know that a radical in mathematics is the opposite of an exponent. Let us recall that for any real number “ a”  and a positive integer “ n”, we define a n  as

a n =  a x a x a x a x a …… ( n times ).

Here a n is called the nth power of a. the real number a is called the base and n is called the exponent of the nth power of a.

Radicals on the other hand are same as the root of a number. In other words, a radical is an expression that involves a root, usually a square root or cube root. The root of a number is expressed using the symbol √, for example, √5, ∛10. etc. The horizontal line covering the number is called the vinculum and the number under it is called the radicand. So, if we have a radical, say $\sqrt[n]{x}$, then it is read as “ x radical n ”. Here, “ n “ is the index and “ x “ is the radicand. Let us now learn about division of radicals.

The division of radicals is done in the same manner as we do for exponents. The general laws of exponents apply for the radicals as well. some of these laws are –

If a and b be positive real numbers, then,

1. ( √a )  2 = a
2. ( √a )  ( √b )  = ( √ab )
3. $\frac{( √a )}{( √b )} = \sqrt{\frac{a}{b}}$
4. (√a + √b ) (√a – √b ) = a – b
5. ( a + √b ) ( a – √b ) = a 2 –  b
6. (√a ± √b ) 2 = a ± 2 $\sqrt{ab}$ + b
7. (√a + √b ) (√c + √d ) = $\sqrt{ac} + \sqrt{ad} + \sqrt{bc} + \sqrt{bd}$

Some other properties of radicals that are important for division are –

1. If two or more radicals are multiplied with the same index, you can take the radical once and multiply the numbers inside the radicals.
This means that n √ a x n √ b = n √ ( a x b )
2. If two radicals are in division with the same index, you can take the radical once and divide the numbers inside the radicals. This means that n √ a ÷ n √ b = n √ ( a ÷  b )
3. One number can be taken out of a square root for every two same numbers multiplied inside the square root. And also, one number can be taken out of a cube root for every three same numbers multiplied inside the cube root and so on. For example, √9 = $\sqrt{3 x 3}$ = 3. Similarly, ∛27 = $\sqrt[3]{3 x 3 x 3}$ = 3
4. A radical with index n can be written as exponent 1/n. This means that $\sqrt[n]{a}$ = a 1/n.

There are two methods of division of radicals. The methods are –

1. Using Quotient Rule of radicals
2. Division through Rationalisation of Denominator

Let us discuss both the methods one by one.

We know that a quotient is the answer to a division problem. The quotient rule states that a radical involving a quotient is equal to the quotients of two radicals. This means that –

$\sqrt[n]{\frac{a}{b}} =\frac{\sqrt[n]a}{\sqrt[n]b}$

In other words, when we divide one radical expression by another, we can simplify it by writing both expressions under the same radical and then simplify them in order to get the result. There are however, some conditions that need to be considered during the application of quotient rule for division of radicals –

1. Each radical has the same index. The index is the superscript number to the left of the radical symbol, which indicates the degree of the radical.

For instance, $\sqrt[4]{16}$ = 2

Here the index is 2 as it is indicating the fourth root of 16. In case of index being mentioned explicitly, the index is treated as 2, i.e. a square root.
1. The denominator of the fraction is not zero.

### Rationalising the denominator

We have learnt that one of the rules for radicals is that for a radical expression to be in its simplest form, it cannot have a radical in the denominator. Sometimes, we come across expressions that have square roots in their denominators. This means that we need to divide a radical expression by a square root of a number. Division of such expressions is convenient if their denominators are free from square roots. To make the denominators free from square roots, we multiply the numerator as well as the denominator by an irrational number. Such a number is called the rationalisation factor. Recall that a number is an irrational number if it has a non-terminating and non-repeating decimal representation. This means that we will simplify any radical expression with a fraction is to make sure that there is not a radical in the denominator.

Let us understand the use of rationalisation for division of radicals through an example.

Example

Suppose we want to find the value of $\frac{1}{2+ √3}$

Solution

We have $\frac{1}{2+√3}$

We know that (√a + √b ) (√a – √b ) = a – b.

Also, ( 2+ √3 ) (2 – √3 ) = 2 2 – ( √3 ) 2 = 4 – 3 = 1

Therefore, in order to remove 3 from the denominator, we will multiply the numerator as well as the denominator by 2- √3. We will have,

$\frac{1}{2+ √3} x \frac{(2- √3 )}{(2- √3 )} = \frac{(2- √3 )}{( 2+ √3 ) (2- √3)} = \frac{(2-√3)}{4-3} = \frac{(2- √3 )}{1} = ( 2- √3 )$

Hence, $\frac{1}{2+\sqrt{3}} = (2-\sqrt{3})$

Also, we can see from the above calculations that the rationalisation factor of $\frac{1}{√a}$ is √a and for $\frac{1}{a ± √b}$, the rationalisation factor is a ∓√b

Let us now summarise the steps involved in the division of radicals.

### Steps involved in the Division of Radicals

The following steps are involved in the division of radicals –

1. The first step is to check the index of the radicals involved and whether the denominator of the fraction is zero. If the index of the radicals is the same and the denominator of the fraction is not zero, we can proceed ahead with the division.
2. The second step is to convert the given expression into one radical.
3. In the third step, we simply the expression to the extent possible.
4. Rationalisation of the denominator, if required is the last step.

Let us understand the above steps through an example.

Example

Solve $\frac{√100}{√4}$

Solution

We have, $\frac{√100}{√4}$

We can see that both the radicals have the same index, i.e. 2. Also, the denominator is not zero. Now, we know that by quotient rule, $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]a}{\sqrt[n]b}$

Therefore, $\frac{√100}{√4} = \frac{√100}{√4} = √25 = 5$

Hence, $\frac{√100}{√4}$ = 5

## Properties of Division of Radicals

Below we have some properties of radicals that are integral to their division as well –

1. If the number is positive under the radical, the result will be positive.
2. If the number is negative under the radical, the result will be negative.
3. If the number under the radical is negative and an index is an even number, the result will be an irrational number.
4. If an index is not mentioned, the radical will be square root.

## Solved Examples

Example 1 If both a and b are rational numbers, find the value of a and b in each of the following equalities –

1. $\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = a + b\sqrt{3}$
2. $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = a – b\sqrt{6}$

Solution a) $\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = a + b\sqrt{3}$

Let us first find the rationalisation factor for the given radical expression. For this we must observe the denominator. The denominator of the given radical expression is 7+4√3. This is of the form a + √b. Now, we have learnt that the rationalisation factor for $\frac{1}{a ± √b}$, the rationalisation factor is a ∓ √b. This means that the rationalisation factor for 7+4√3 will be 7-4√3. Therefore, we will multiply the numerator and the denominator of the given radical expression by 7-4√3. Hence, multiplying the numerator and the denominator of the given radical expression by 7-4√3, we get

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\:=\:\frac{5+2\sqrt{3}}{7+4\sqrt{3}} x \frac{7-4\sqrt{3}}{7-4\sqrt{3}} = \frac{(5+2\sqrt{3})(7-4\sqrt{3})}{(7+4\sqrt{3})(7-4\sqrt{3})}$

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = \frac{5 x 7-5 x 4\sqrt{3}+2\sqrt{3} x 7-2\sqrt{3} x 4\sqrt{3}}{7^2-( 4\sqrt{3})^2}$

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = \frac{35-20\sqrt{3}+14\sqrt{3} – 8\sqrt{3×3}}{49-16x\sqrt{3}^2}$

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = \frac{35-(20-14)\sqrt{3}- 8\sqrt{3×3}}{49-16\:x\:3}$

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = \frac{35-(20-14)\sqrt{3}-8×3}{49-48} = \frac{36-6\sqrt{3}-24}{49-48} = 11-6\sqrt{3}$

Hence, $\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = 11-6\sqrt{3}$ ………… ( 1 )

Now, we have been given that $\frac{5+2\sqrt{3}}{7+4\sqrt{3}} = a+b\sqrt{3}$ ……… ( 2 )

From equation ( 1 ) and equation (2 ), we have that

a + b√3 = 11 – 6√3

Comparing the coefficients on both the sides, we get

a = 11 and b = -6

Hence, if $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$ = a + b√3, then a = 11 and b = -6

b) $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}$ = a  – b√6

Let us first find the rationalisation factor for the given radical expression. For this we must observe the denominator. The denominator of the given radical expression is 3√2-2√3. This is of the form √a – √b. Now, we have learnt that the rationalisation factor for $\frac{1}{a ±√b}$, the rationalisation factor is a ∓√b. This means that the rationalisation factor for 3√2-2√3.  will be 3√2+2√3. Therefore, we will multiply the numerator and the denominator of the given radical expression by 3√2+2√3. Hence, multiplying the numerator and the denominator of the given radical expression by 3√2+2√3 we get

$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} x \frac{3\sqrt{2}+2\sqrt{3}}{3\sqrt{2}+2\sqrt{3}}$

⇒$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{(\sqrt{2}+\sqrt{3})(3\sqrt{2}+2\sqrt{3})}{(3\sqrt{2}-2\sqrt{3})(3\sqrt{2}+2\sqrt{3})}$

⇒$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{\sqrt{2}\:x\:3 \sqrt{2}+ \sqrt{2}\:x\:2\sqrt{3}+\sqrt{3}\:x\:3\sqrt{2} +\sqrt{3}\:x\:2\sqrt{3}}{( 3\sqrt{2})^2-(2\sqrt{3})^2}$

⇒$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{3\sqrt{2 x 2}+2\sqrt{3 x 2} +3\sqrt{3 x 2}+2\sqrt{3 x 3}}{9 (\sqrt{2})^2-4 (\sqrt{3})^2}$

⇒$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{3×2+2\sqrt{6}+3\sqrt{6}+2×3}{9\:x\:2-4\:x\:3}$

⇒$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = \frac{6+(2+3)\sqrt{6}+6}{18-12}=\frac{12+5\sqrt{6}}{6}=2+\frac{5}{6}\sqrt{6}$

Hence, $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = 2+\frac{5}{6}\sqrt{6}$ ……………… ( 1 )

Now, we have been given that $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} = a-b\sqrt{6}$ …….. ( 2 )

From equation ( 1 ) and equation (2 ), we have that

$a – b\sqrt{6} = 2 + \frac{5}{6}\sqrt{6}$

Comparing the coefficients on both the sides, we get

a = 2 and b = -$\frac{5}{6}$

Hence, if $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}$ = a  – b√6, then a = 2 and b = -$\frac{5}{6}$

Example 2 Divide √80 by √10

Solution We need to divide √80 by √10. This means that we need to find the value of √80 ÷ √10. This radical expression can also be written as $\frac{√80}{√10}$.

Therefore, we need to find the value of $\frac{√80}{√10}$

Now, we can see that both the radicals have the same index, i.e. 2. Also, the denominator is not zero. Now, we know that by quotient rule, $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Therefore, $\frac{√80}{√10} = \sqrt{\frac{80}{10}} = \sqrt{\frac{8 x 10}{10}} = \sqrt{8}$

Now, we know that 8 can further be factorised as 8 = 2 x 2 x 2

This means that we can further write $\sqrt{8} = \sqrt{2 x 2 x 2} = 2\sqrt{2}$

Hence, √80 divided by √10 will result in 2√2.

Example 3 Divide √81 by √25

Solution We need to divide √81 by √25. This means that we need to find the value of √81 ÷ √25. This radical expression can also be written as $\frac{81}{25}$.

Now, we can see that both the radicals have the same index, i.e. 2. Also, the denominator is not zero. Now, we know that by quotient rule, $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Therefore, $\frac{\sqrt{81}}{\sqrt{25}} = \sqrt{\frac{81}{25}} = \sqrt{\frac{9 x 9}{5 x 5}} = \frac{9}{5}$

Hence, the division of √81 by √25  will result in $\frac{9}{5}$

## Key Facts and Summary

1. A radical is an expression that involves a root, usually a square root or cube root.
2. The root of a number is expressed using the symbol √, for example, √5, ∛10. etc. The horizontal line covering the number is called the vinculum and the number under it is called the radicand.
3. ( √a )  2 = a
4. ( √a )  ( √b )  = ( √ab )
5. $\frac{( √a )}{( √b )} = \sqrt{\frac{a}{b}}$
6. (√a + √b ) (√a – √b ) = a – b
7. ( a + √b ) ( a – √b ) = a 2 –  b
8. (√a ± √b ) 2 = a ± 2 $\sqrt{ab}$ + b
9. (√a + √b ) (√c + √d ) = $\sqrt{ac} + \sqrt{ad} + \sqrt{bc} + \sqrt{bd}$
10. If two or more radicals are multiplied with the same index, you can take the radical once and multiply the numbers inside the radicals.
This means that n √ a x n √ b = n √ ( a x b )
11. If two radicals are in division with the same index, you can take the radical once and divide the numbers inside the radicals. This means that n √ a ÷ n √ b = n √ ( a ÷  b )
12. One number can be taken out of a square root for every two same numbers multiplied inside the square root. And also, one number can be taken out of a cube root for every three same numbers multiplied inside the cube root and so on. For example, $\sqrt{9} = \sqrt{3 x 3} = 3$. Similarly, ∛27 = $\sqrt[3]{3 x 3 x 3} = 3$
13. A radical with index n can be written as exponent 1/n. This means that $\sqrt[n]{a}$ = a 1/n.
14. If the number is positive under the radical, the result will be positive.
15. If the number is negative under the radical, the result will be negative.
16. If the number under the radical is negative and an index is an even number, the result will be an irrational number.
17. If an index is not mentioned, the radical will be square root.