If m and n are natural numbers such that √(7+√48) = √m + √n then m² +
n² equals

a)25
b)37
c)29
d)40

Step by step answer please

25

Step-by-step explanation:

√(7+√48) = √(4+3+√(16.3) ) =√(2² + (√3)² + 4√3)) =√(2² + (√3)² + 2.2.√3))

= √(2 + √3)² = 2 + √3

Therefore from (1),

2 + √3 = √m + √n

There is a theorem on surd which states that if √(a+√b) = √x + √y , then √(a-√b) = √x-√y . Application of the theorem to the above equation leads us to the following two linear algebraic equations in the two unknown variables m and n.

2 + √3 = √m + √n

2 - √3 = √m - √n

Add and subtract the above two equations to obtain

2 = √m and √n = √3 These give on squaring,

m = 4 and n = 3 Substituting the above values for m and n,

m² + n² = 4² + 3² = 16 + 9 = 25