Rationalize the denominator: 5 2 - 3

Given expression: 5 2 - 3 The rationalizing factor of 2 - 3 is 2 + 3 . We multiply the numerator and denominator by 2 + 3 . So, 5 2 - 3 = 5 × 2 + 3 2 - 3 × 2 + 3 = 10 + 5 3 2 2 - 3 2 : Using the identity: a 2 - b 2 = a - b a + b = 10 + 5 3 4 - 3 = 10 + 5 3 1 = 10 + 5 3

Rationalize the denominator: 5 2 - 3

Given expression: 5 2 - 3 The rationalizing factor of 2 - 3 is 2 + 3 . We multiply the numerator and denominator by 2 + 3 . So, 5 2 - 3 = 5 × 2 + 3 2 - 3 × 2 + 3 = 10 + 5 3 2 2 - 3 2 : Using the identity: a 2 - b 2 = a - b a + b = 10 + 5 3 4 - 3 = 10 + 5 3 1 = 10 + 5 3

Rationalize the denominator: 3 2 + 3

Given expression: 3 2 + 3 The rationalizing factor of 2 + 3 is 2 - 3 . We multiply the numerator and denominator by 2 - 3 . So, 3 2 + 3 = 3 × 2 - 3 2 + 3 × 2 - 3 = 3 2 - 3 2 2 - 3 2 = 3 2 - 3 2 - 3 = 3 2 - 3 - 1 Hence, the required answer is - 3 2 - 3 = 3 3 - 2 .

Rationalize the denominator: 4 5 - 3

Given expression: 4 5 - 3 The rationalizing factor of 5 - 3 is 5 + 3 . We multiply the numerator and denominator by 5 + 3 . So, 4 5 - 3 = 4 × 5 + 3 5 - 3 × 5 + 3 = 4 5 + 3 5 2 - 3 2 : Using the identity a 2 - b 2 = a - b a + b = 4 5 + 3 5 - 3 = 4 5 + 3 2 = 2 5 + 3 Hence, the required answer is 2 5 + 3 .

Write 3 irrational numbers between. 2 and 7

Given numbers are 2 and 7 . Let us find the squares of the given numbers. 2 = 2 2 = 2 , 7 = 7 2 = 7 We need 3 numbers whose squares are between 2 and 7 . We can take the numbers 3 , 5 and 6 . We know that 7 > 6 > 5 > 3 > 2 So, 7 > 6 > 5 > 3 > 2 The required numbers are 6 , 5 , 3 .

Write 3 irrational numbers between. 15 and 20

Given numbers are 15 and 20 . Let us find the squares of the given numbers. 15 = 15 2 = 15 , 20 = 20 2 = 20 We need 3 numbers whose squares are between 15 and 20 . We can take the numbers 17 , 18 , and 19 . We know that 20 > 19 > 18 > 17 > 15 So, 20 > 19 > 18 > 17 > 15 The required numbers are 19 , 18 , 17 .