# HOW TO EXPRESS A GIVEN SURD TO SIMPLEST FORM

The following steps will be useful to reduce the given surd to its simplest form.

Step 1 :

Find factors of the number inside the given radical.

Step 2 :

Based on the order of the radical, we have to group them as pairs and factor out.

Step 3 :

In case we already have terms outside the radical, we have to multiply them by the factor taken out and multiply the remaining numbers inside the radical.

Express the following surds in simplest form.

Example 1 :

332

Solution :

332  =  3√(2 ⋅ ⋅ 2 ⋅ 2 ⋅ 2)

Order of the given radical is 3.

So, we have to factor out one term for every three same terms.

=  23√(2 ⋅ 2)

=  23√4

Example 2 :

√63

Solution :

√63  =  √(3 ⋅ 3 ⋅ 7)

Order of the given radical is 2.

So, we have to factor out one term for every two same terms.

=  3√7

Example 3 :

√243

Solution :

√243  =  √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3)

Order of the given radical is 2.

So, we have to factor out one term for every two same terms.

=  (3 ⋅ 3) √3

=  9√3

Example 4 :

3√256

Solution :

3√256  =  3√(2 ⋅ ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2)

Order of the given radical is 3.

So, we have to factor out one term for every three same terms.

=  2 ⋅ 2 ⋅ 2 ⋅ 2

=  16

Example 5 :

4√80

Solution :

4√80  =   4√(2 ⋅ 2 ⋅ 2  ⋅ 2 ⋅ 5)

Order of the given radical is 4.

So, we have to factor out one term for every four same terms.

=  245

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