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## Some General Rules of Radical Expressions

Some general rules of the radical functions are given below:

• The result of the radical operation is a non-negative, i.e. a positive number if the number inside the radical symbol is positive.
• The result of the radical operation is negative if the number inside the radical symbol is negative or the index of the expression is an odd number.
• If the number inside the radical symbol is negative and the index or exponent of the expression is an even number, then the result of the expression is an irrational number.

Two radical expressions can be added or subtracted only if they have the same index and the radicand. For instance, consider the following three radicals with same index k and the radicand a:

In the next section, we will solve a couple of examples in which we will add the radical expressions.

## Example 1

Simplify

### Solution

To add the above radicals, first, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is 4 and the radicand is 5. Hence, we know that we can add these radicals.

=

=

=

## Example 2

Simplify

### Solution

To add the above radicals, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is 3 and the radicand is 4. Hence, we know that we can add these radicals.

=

=

=

## Example 3

Simplify

### Solution

To add the above radicals, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is the same, i.e. 3, however the radicands are different. In this case, we will see whether we can make the radicands the same or not.

can be written as . Simplifying it further will give us the following expression:

can be written as . Simplifying it further will give us the following expression:

Now, all three radical expressions have the same radicand and same index, so we can easily add them:

## Example 4

Simplify

### Solution

To add the above radicals, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is the same, i.e. 2, however the radicands are different. In this case, we will see whether we can make the radicands the same or not.

can be written as . Simplifying it further will give us the following expression:

can be written as . Simplifying it further will give us the following expression:

Now, all three radical expressions have the same radicand and same index, so we can easily add them:

## Example 5

Simplify

### Solution

To add the above radicals, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is the same, i.e. 4, however the radicands are different. In this case, we will see whether we can make the radicands the same or not.

can be written as . Simplifying it further will give us the following expression:

can be written as . Simplifying it further will give us the following expression:

can be written as . Simplifying it further will give us the following expression:

Now, all three radical expressions have the same radicand and same index, so we can easily add them:

## Example 6

Simplify

### Solution

To add the above radicals, we will see whether the index and the radicands are the same or not. In the above example, the index of all three radical expressions is the same, i.e. 3, however the radicands are different. In this case, we will see whether we can make the radicands the same or not.

can be written as . Simplifying it further will give us the following expression:

can be written as . Simplifying it further will give us the following expression:

Now, all three radical expressions have the same radicand and same index, so we can easily add them:

## Example 7

Simplify

### Solution

Take the L.C.M of the numbers in the denominator. The L.C.M of 5 and 2 is 10. Hence, we can write it as:

## Example 8

Simplify .

### Solution

First, simplify the above expression like this:

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