In each of the following determine rational numbers a and b.
(i) √3-1/√3+1= a-b√3 (ii)4+√2/2+√2= a-√b
(iii) 3 + √2 / 3-√2 = a+b √2
(iv) 5 + 3√3 / 7 + 4√3 = a+b√3 (v) √11 - √7 / √11 + √7 = a-b √77 (vi) 4 + 3√5 / 4 - 3√5 = a + b√5
Answers
Given : (i) √3 - 1/√3 + 1 = a - b√3
The Rationalisation factor for √3 + 1 is √3 - 1 . Therefore , multiplying and dividing by √3 - 1 we have :
= (√3 - 1)(√3 - 1) /(√3 + 1)(√3 - 1)
= (√3 - 1)²/√3² - 1²
By Using Identity : (a - b)² = a² + b² - 2ab & (a + b)(a – b) = a² - b²
= [√3² + 1² - 2 ×√3 × 1 ]/ (3 - 1)
= [3 + 1 - 2√3]/2
= [4 - 2√3]/2
= 2 - √3
We have , √3 - 1/√3 + 1 = a - b√3
2 - √3 = a - b√3
On equating rational and Irrational parts we obtain :
a = 2 , b = 1
Hence rational numbers a is 2 and b is 1.
(ii) 4 + √2/2 + √2 = a - √b
The Rationalisation factor for 2 + √2 is 2 - √2 . Therefore , multiplying and dividing by 2 - √2 we have :
= (4 + √2)(2 - √2) / (2 + √2) (2 - √2)
= [2(4 + √2) - √2 (4 + √2)]/ (2)² - (√2)²
= [8 + 2√2 - 4√2 - √2 √2]/(4 - 2)
= [8 - 2√2 - 2]/ 2
= (6 - 2√2)/2
= 3 - √2
We have , 4 + √2/2 + √2 = a - √b
3 - √2 = a - √b
On equating rational and Irrational parts we obtain :
a = 3 , b = 2
Hence rational numbers a is 3 and b is 2.
(iii) 3 + √2 / 3 -√2 = a + b √2
The Rationalisation factor for 3 - √2 is 3 + √2 . Therefore , multiplying and dividing by 3 + √2 we have :
= (3 + √2)(3 + √2) / (3 - √2) (3 + √2)
By Using Identity : (a + b)² = a² + b² + 2ab & (a + b)(a – b) = a² - b²
= [(3 + √2)²]/ (3)² - (√2)²
= [3² + √2² - 2 × 3 × √2 ]/ (9 -2)
= [9 + 2 - 6√2]/ 7
= (11 - 6√2]/ 7
= 11/7 - 6√2/ 7
We have , 3 + √2 / 3 -√2 = a + b √2
11/7 - 6√2/ 7 = a + b √2
On equating rational and Irrational parts we obtain :
a = 11/7, b = 6/7
Hence rational numbers a is 11/7 and b is 6/7.
(iv) 5 + 3√3 / 7 + 4√3 = a + b√3
The Rationalisation factor for 7 + 4√3 is 7 - 4√3 . Therefore , multiplying and dividing by 7 - 4√3 we have :
= (5 + 3√3) (7 - 4√3) / (7 + 4√3)(7 - 4√3)
= [7(5 + 3√3) - 4√3(5 + 3√3)]/ (7)² - ( 4√3)²
= [35 + 21√3 - 20√3 - 12 3]/(49 - 16 × 3)
= [35 + √3 - 36]/(49 - 48)
= [35 - 36 + √3]/(1)
= √3 - 1
We have , 5 + 3√3 / 7 + 4√3 = a + b√3
√3 - 1 = a + b√3
On equating rational and Irrational parts we obtain :
a = -1 , b = 1
Hence rational numbers a is -1 and b is 1.
(v) √11 - √7 / √11 + √7 = a - b √77
The Rationalisation factor for √11 + √7 is √11 - √7 . Therefore , multiplying and dividing by √11 - √7 we have :
= (√11 - √7)(√11 - √7) / (√11 + √7 )(√11 + √7)
= (√11 - √7)²/√11² - √7²
By Using Identity : (a - b)² = a² + b² - 2ab & (a + b)(a – b) = a² - b²
= [√11² + √7² - 2 ×√11 × √7 ]/ (11 - 7)
= [11 + 7 - 2√77]/4
= [18 - 2√77]/4
= 2[9 - √77]/4
= 9 - √77]/2
= 9/2 - √77/2
We have , √11 - √7 / √11 + √7 = a - b √77
9/2 - √77/2 = a - b √77
On equating rational and Irrational parts we obtain :
a = 9/2 , b = 1/2
Hence rational numbers a is 9/2 and b is 1/2.
(vi) 4 + 3√5 / 4 - 3√5 = a + b√5
The Rationalisation factor for 4 - 3√5 is 4 + 3√5 . Therefore , multiplying and dividing by 4 + 3√5 we have :
= (4 + 3√5)( 4 + 3√5) / (4 - 3√5)(4 + 3√5)
= (4 + 3√5)² /(4)² - (3√5)²
By Using Identity : (a + b)² = a² + b² + 2ab & (a + b)(a – b) = a² - b²
= [4² + 3√5² + 2 × 4 × 3√5 ] / (16 - 9 × 5)
= [16 + 9 × 5 + 24√5]/(16 - 45)
= [16 + 45 + 24√5]/( -29)
= (61 + 24√5) /(-29)
= - (61 + 24√5)/ 29
= - 61/29 - 24√5/ 29
We have , 4 + 3√5 / 4 - 3√5 = a + b√5
- 61/29 - 24√5/ 29 = a + b√5
On equating rational and Irrational parts we obtain :
a = - 61/29 , b = - 24/29
Hence rational numbers a is - 61/29 and b is - 24/29.
HOPE THIS ANSWER WILL HELP YOU…..
Some questions of this chapter :
Simplify the following expressions :
(i) (11+√11) (11-√11)(ii) (5+√7) (5-√7)
(iii) (√8 - √2) (√8 - √2)
(iv) (3+√3) (3-√3)(v) (√5 - √2) (√5 - √2)
https://brainly.in/question/15897354
Find the value to three places of decimals of each of the following. It is given that
√2= 1.414,√3 = 1.732,√5 =2.236 and √10 = 3.162.
(i)2/√3 (ii)3/√10 (iii) √5 + 1 / √2 (iv) √10 + √15 / √2
(v) 2+ /√3 / 3 (vi) √2 - 1 / √5
https://brainly.in/question/15897357
Answer:
Step-by-step explanation:
⠀⠀⠀⠀ 3 + √2 / 3 -√2 = a + b √2
The Rationalisation factor for 3 - √2 is 3 + √2 . Therefore , multiplying and dividing by 3 + √2 we have :
= (3 + √2)(3 + √2) / (3 - √2) (3 + √2)
By Using Identity : (a + b)² = a² + b² + 2ab & (a + b)(a – b) = a² - b²
= [(3 + √2)²]/ (3)² - (√2)²
= [3² + √2² - 2 × 3 × √2 ]/ (9 -2)
= [9 + 2 - 6√2]/ 7
= (11 - 6√2]/ 7
= 11/7 - 6√2/ 7
We have , 3 + √2 / 3 -√2 = a + b √2
11/7 - 6√2/ 7 = a + b ⠀⠀⠀⠀⠀⠀⠀