Irrational Numbers Class 9

 

Class 9

IRRATIONAL Numbers : If  a number cannot be expressed in the form p/q, where p,q, (q ≠ 0) are integer and do not have any common factor except 1, then it is an irrational number.

1. Express the following surds in simplest form.

i)                 √ 75 = √3*5*5 = 5√3

ii)                √ 80 = √4*4*5 = 4√5

iii)              √ 96 = √4*4*6 = 4√6

iv)              √ 112 = √4*4*7 = 4√7

v)                √ 162 = √9*9*2 = 9√2 

2.  Write the simplest rationalising factor of:

i)                 √32 = √2*2*2*2*2= 4√2

                   so here rationalising factor of √32 is √2

ii)                √48 = √4*4*3 = 4√3

                   so here rationalising factor of √48 is √3

 

iii)              √72 = √3*3*2*2*2 = 6√2

                    so here rationalising factor of √72 is √2

 

iv)              5√3 + 8

   To rationalise 5√3 + 8, multiply by 5√3 - 8

   so here rationalising factor of 5√3  + 8 is 5√3 - 8

 

v)                √7 -√5

To rationalise (√7 -√5), multiply by (√7 +√5)

so here rationalising factor of √7 -√5 is √7 +√5

 

vi)              √6 + √3

To rationalise (√6 + √3), multiply by (√6 - √3)

so here rationalising factor of √6 + √3 is √6 - √3

 

 

3. Simplify the following:

i) √75 - √12 + √27

=√5*5*3 - √2*2*3 + √ 3*3*3

= 5√3 - 2√3 + 3√3

= (5 – 2 +3) √3

= 6√3

ii) 3√11 + 8√99 - √44

= 3√11 + 8√ 9*11 - √4*11

= 3√11 + 8*3√11 - 2√11

= 3√11 + 24√11 - 2√11

= (3+24-2) √11

= 25√11

iii) 3√63 - 2√28 - √112

= 3√9*7 - 2√4*7 - √16*7

= 3*3√7 – 2*2√7 - 4√7

= 9√7 - 4√7 - 4√7

= (9-4-4) √7

= 1√7

= √7

iv) 5√18 +7√50 - 10√8

= 5√2*9 + 7√25*2 - 10√4*2

= 5*3√2 + 7*5 √2 – 10*2√2

= 15√2 + 35√2 - 20√2

= (15 +35 -20) √2

= 30√2

v) 6√72 - √18 - 8√32

= 6√36*2 - √9*2 - 8√16*2

= 6*6√2 - 3√2 - 8*4√2

= 36√2 - 3√2 -32√2

= (36 - 3 - 32) √2

= 1√2 = √2

4. Expand:

i)  ( 2 + √5)² = 2² + (2*2*√5) +(√5)²                                          using (a + b)² = a² + 2ab + b²       

                     = 4 + 4√5+ 5

                     =9+4√5

ii) (7 - √3)² = 7² - (2*7*√3) +(√3)²                                             using (a - b)² = a² - 2ab + b²       

                  = 49 - 14√3+ 3

                   = 52 - 14√3

iii) (2√3 - 5√2)² = (2√3)² - (2*2√3*5√2) +(5√2)²                         using (a - b)² = a² - 2ab + b²       

                        = 12 - 20√6+ 50

                        = 62 - 20√6

iv) (4√5 + 3√7)² = (4√5)² + (2*4√5*3√7) + (3√7)²                       using (a + b)² = a² + 2ab + b²       

                        = 80 + 24√35 + 63

                        = 143 + 24√35

5) Find the Product

i) (3 - √5) (3 + √5) = 3² - √5² = 9 – 5 = 4                                using (a+b) (a-b) = a² - b²

ii) (√2 + 3√11) (5√2 - 2√11)

  =√2*5√2 - √2*2√11 + 3*5*√11*2 – 3*2*11

  =10 - 2√22 + 15√22 - 66

  = 13√22 - 56

iii) (5√7 + 3) (4 - √7)

= 5√7*4 -5√7*√7 +3*4 – 3*√7

= 20√7 – 35 + 12 - 3√7

= 17√7 - 23