Question

What is the square root of 16 +6√7 is

Answer 1

Answer:

√(16 +6√7) = 3 + √7

Step-by-step explanation:

Let say square root of  16 +6√7    is  a + b√c

=> a + b√c  = √(16 +6√7)

Squaring both sides

=> a² + b²c  + 2ab√c  = 16 +6√7

Comparing

2ab√c = 6√7

=> c = 7

=> 2ab = 6

=> ab = 3

=> b = 3/a

a² + b²c = 16

=> a² + 7b² = 16

=> a²  + 7(3/a)² = 16

=> a⁴  - 16a²  + 63 = 0

=> a⁴ - 7a² - 9a² + 63 = 0

=> (a² - 7)(a² - 9) = 0

=> a = √7   a = 3

=> b =  3/√7   b = 1

using a = 3 & b = 1

a + b√c =  3 + 1√7 = 3  + √7

using a = √7 & b = 3/√7

or √7 + (3/√7)√7 =  √7 + 3  = 3 + √7

Hence √(16 +6√7) = 3 + √7

another method

16 + 6√7

= 7 + 9 + 6√7

= 7 + 3√7 + 3√7  + 9

= √7 (√7 + 3) + 3(√7 + 3)

= (√7 + 3)(√7 + 3)

=(√7 + 3)²

= (3 + √7)²

=> √(16 +6√7) = 3 + √7