HOW TO EXPRESS A GIVEN SURD TO SIMPLEST FORM

Express the following surds in simplest form.

Example 1 :

³√32

Solution :

³√32  =  ³√(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.

  =  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 ⋅ 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.

  =  2⁴√5

Example 6 :

(2√3 x 2√6)/4

Solution :

= (2√3 x 2√6)/4

Using the property √a x √b = √(a x b)

= (4√(3x6)/4

= (4√(3 x 3 x 2)/4

= 3√2

Example 7 :

(√12 x √27) / (8 x 2√6)

Solution :

= (√12 x √27) / (8 x 2√6)

Using the property √a x √b = √(a x b)

= (√(12 x 27) / (8 x 2√6)

= (√(2⋅2⋅3⋅3⋅3⋅3) / (8 x 2√6)

= 12 / 16√6

= 3/4√6

Rationalizing the denominator, we get

= (3/4√6) ⋅ (√6/√6)

= 3√6/4(6)

= 3√6/24

= √6/8

Example 8 :

(√2 + 3) (√2 - 5) 

Solution :

= (√2 + 3) (√2 - 5) 

Using distributive property, we get

= √2 √2 + √2(-5) + 3√2 - 15

= 2 - 5 √2 + 3√2 - 15

= - 2√2 - 13

Example 9 :

(√2 + 1)²

Solution :

= (√2 + 1)²

Using the algebraic identity, 

 (a + b)² = a²+ 2ab + b²

 (√2 + 1)² = √2²+ 2√2(1) + 1²

= 2 + 2√2 + 1

= 3 + 2√2

Example 10 :

2√3 + 4√3

Solution :

2√3 + 4√3

Inside the radical, since we have same values these can be considered as like terms. Combining the like terms, we get

= 6 √3

Example 11 :

√27 + √3

Solution :

= √27 + √3

√27 = √(3 ⋅ 3 ⋅ 3)

= 3√3

Example 12 :

√57 ÷ √3

Solution :

= √57 ÷ √3

= √(3⋅19) ÷ √3 

= √3 √19 ÷ √3

= √19

Example 13 :

√38 ÷ √2

Solution :

= √38 ÷ √2

= √(2⋅19) ÷ √2 

= √2 √19 ÷ √2

= √19

Example 14 :

5√90

Solution :

= 5√90

= 5√(2⋅5⋅3⋅3)

= (5 ⋅ 3) √(2⋅5)

= 15 √10

Example 15 :

2√75 - √45 + 2√20

Solution :

= 2√75 - √45 + 2√20

Decomposing into prime factors, we get

= 2√(3⋅5⋅5) - √(3⋅3⋅5) + 2√(2⋅2⋅5)

= (2⋅5)√3 - 3√5 + (2⋅2)√5

= 6√3 - 3√5 + 4√5

Combining the like terms, we get

= 6√3 + √5 

Example 16 :

(3√2)²

Solution :

= (3√2)²

Distributing the power, we get

= 3²√2²

= 9(2)

= 18

Example 17 :

(√18 x √2)

Solution :

= (√18 x √2)

Decomposing into prime factors, we get

= (√(3⋅3⋅2) x √2)

= √(3⋅3⋅2⋅2)

= 6

Example 18 :

2√3(3√2 - √3)

Solution :

= 2√3(3√2 - √3)

Using distributive property, we get

= 3√2(2√3) - √3 (2√3)

= 6√6 - 2(3)

= 6√6 - 6

Example 19 :

(√7 - √3)²

Solution :

= (√7 - √3)²

Using algebraic identity, we get

 (a - b)² = a² - 2ab + b²

= (√7)² - 2√7√3 - (√3)²

= 7 - 2√(7x3) - 3

= 4 - 2√21

Example 20 :

(2√m + 5)²

Solution :

= (2√m + 5)²

Using algebraic identity, we get

 (a + b)² = a² + 2ab + b²

= (2√m)² - 2(2√m)(5) - 5²

= 4m - 10√m - 25

It cannot be simplified further, so the answer is

4m - 10√m - 25

Example 21 :

(5√2 - 3√3)(5√2 + 3√3)

Solution :

= (5√2 - 3√3)(5√2 + 3√3)

It exactly matches with algebraic identity

(a + b)(a - b) = a² - b²

Here a = 5√2 and b = 3√3

= (5√2)²(3√3)²

= 25(2) (3(3))

= 50 (9)

= 450

Example 22 :

(2 + √3)/(2√3)

Solution :

= (2 + √3)/(2√3)

= [(2 + √3)/(2√3)][(2√3)/(2√3)]

(2√3)(2+ √3)/4(3)

(2√3)(2+ √3)/12

= √3(2+ √3)/6

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