**Simplify radical expressions with variables :**

Here we are going to learn how to simplify radical expressions with variables.

To simplify radical expression we need to follow the steps given below.

**Step 1 :**

Find the factors of the number and variable inside the radical sign.

**Step 2 :**

According to the index of the radical we can take one term from the radical sign.

- If we have square root (√), we have to take only one common term for every two same terms.
- If we have cube root (∛), we have to take only one common term for every three same terms.
- If we have cube root (∜), we have to take only one common term for every four same terms.

Let us see some examples based on the concept.

**Example 1 :**

Simplifying the following radical expressions

**Solution :**

** = **√(2 x 2 x 2 x 2 x u x u x u x u x v x v x v)

Index of the given radical is 2.

Here we have four "2", four "u" and three "v". According to the index, we have to take two 2's, one u and one v instead of 2 same terms from the radical sign.

So we get

= 2 x 2 x u x u x v** **√v

= 4 u²v √v

Hence the answer is 4 u²v √v.

**Example 2 :**

Simplifying the following radical expressions

**Solution :**

** = **√(3 x 7 x 7 x m x m x m x n x n x n)

Index of the given radical is 2.

Number of "3" that we have = 1

Number of "7" that we have = 2

number of "m" that we have = 3 and

number of "n" that we have = 3

According to the index of the radical, we can take one term from the radical for every two same terms . So, we have to take one "7", one "m" and one "n" from the radical.

Hence the answer is 7mn √3mn

**Example 3 :**

Simplifying the following radical expressions

**Solution :**

** = **√(3 x 5 x 5 x X x X x y)

Index of the given radical is 2.

Number of "3" that we have = 1

Number of "5" that we have = 2

number of "x" that we have = 2 and

number of "y" that we have = 1

According to the index of the radical, we can take one term from the radical for every two same terms. So, we have to take one "5", one "x" from the radical.

Hence the answer is 5x √3y

**Example 4 :**

Simplifying the following radical expressions

**Solution :**

** = 6 **√(2 x 6 x 6 x X x X)

Index of the given radical is 2.

Number of "2" that we have = 1

Number of "6" that we have = 2

number of "x" that we have = 2 and

According to the index of the radical, we can take one term from the radical for every two same terms. So, we have to take one "6", one "x" from the radical.

If we already have any number or variable out side radical sign, then we have to multiply those terms with the new terms.

= 6 (6x)√2 = 36 √2

Hence the answer is 36 √2.

After having gone through the stuff given above, we hope that the students would have understood "Simplify radical expressions with variables".

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