# MULTIPLICATION OF SURDS HAVING DIFFERENT ORDERS

## About "Multiplication of surds having different orders"

Multiplication of surds having different orders :

To find the product of two radical terms with different orders, we have to convert both radical terms with same order.

The following example will illustrate the process of finding product of surds having different orders.

Example 1 :

Simplify the following

3   6√5

Solution :

The order of the first radical term is 3 and the order of the second radical term is 6.

Since the orders of both radical terms are not same, we have to find L.C.M for 3 and 6.

L.C.M  =  6

Now we have to convert the first radical term of order 3 as the radical term of order 6.

In order to do this conversion, we have to multiply the order by 2 and express the number which is inside the radical as the product of two same terms.

3 =  3 (2)√(⋅ 3)  =  6√9

6√5  =  6√5

3   6√5  =  6√9 ⋅ 6√5

=  6√(9 ⋅ 5)

=  6√45

Example 2 :

Simplify the following

4√5    √3

Solution :

The order of the first radical term is 4 and the order of the second radical term is 2.

Since the orders of both radical terms are not same, we have to find L.C.M for 4 and 2.

L.C.M  =  4

Now we have to convert the second radical term of order  2 as the radical term of order 4.

In order to do this conversion, we have to multiply the order by 2 and express the number which is inside the radical as the product of two same terms.

√3  =  (2  2)√(3  3)  =  4√9

4√5  =  4√5

4√5    √3  =  4√5 ⋅ 4√9

=  4√(5 ⋅ 9)

=  4√45

Example 3 :

Simplify the following

3√4    4√5

Solution :

The order of the first radical term is 3 and the order of the second radical term is 4.

Since the orders of both radical terms are not same, we have to find L.C.M for 3 and 4.

L.C.M  =  12

Now we have to convert both radical terms of order 12.

3√4  =  (3 ⋅ 4)√(4 ⋅ 4 ⋅ 4 ⋅ 4)  =  12√256

4√5  =  (4 ⋅ 3)√(5 ⋅ 5 ⋅ 5)  =  12√125

3√4 ⋅ 4√5  =  12√256  12√125

=  12√(256 ⋅ 125)

=  12√32000

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