Question

9+2√14,Find the square roots of following surd.

Answer 1

HELLO DEAR,

We know that

(√a+√b)² = a + b + 2√ab

Now √(9 + 2√14). If it is a perfect sq,

so, there will be 2 numbers

such that:-
a + b = 9---------( 1 )

2√ab = 2√14 ⇒ ab = 14

using formula:- (a - b)² = (a + b)² - 4ab

(a - b)² = (9)² - 4(14)

(a - b)² = 81 - 56

(a - b)² = 25

(a - b) = 5--------( 2 )

from------( 1 ) & --------( 2 )

a + b = 9
a - b = 5
–—–—–
2a = 14

a = 7 [ put in ---( 1 ) ]

a + b = 9

b = 9 - 7

b = 2

thus, a = 7 , b = 2

therefore, √(9 + 2√14) = √(√7 + √2)²

= (√7 + √2)

I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Hi ,

√9 + 2√14

= √ ( 7 + 2 ) + 2√( 7 × 2 )

= √ [ ( √7 )² + ( √2 )² + 2 × √7 × √2 ]

= √ ( √7 + √2 )²

= √7 + √2

Therefore ,

√ 9 + 2√14 = √7 + √2

I hope this helps you.

: )