Each side of a cube is exactly (1+√2)m long. Find exact volume, giving the answer in the form (a+b√2) where a and b are integers.?

V = (1+√2)³
= 1 + 3√2 + 3√2² + √2³
= 1 + 3√2 + 3·2 + 2√2
= 1 + 3√2 + 6 + 2√2
= 7 + 5√2

　
Binomial expansion:

(x+y)³ = x³ + 3x²y + 3xy² + y³
(1+y)³ = 1 + 3y + 3y² + y³

(1 + √2)³ = 1 + 3(√2) + 3(√2)² + (√2)³
(1 + √2)³ = 1 + 3√2 + 6 + 2√2
(1 + √2)³ = 7 + 5√2

Or you could expand using distributive property:

(1 + √2)³
= (1 + √2) (1 + √2) (1 + √2)³
= (1 + √2) [1 (1 + √2) + √2 (1 + √2)]
= (1 + √2) (1 + √2 + √2 + 2)
= (1 + √2) (3 + 2√2)
= 1 (3 + 2√2) + √2 (3 + 2√2)
= 3 + 2√2 + 3√2 + 4
= 7 + 5√2

[ 1 + √2 ] [ 1 + √2 ] [ 1 + √2 ]
[ 1 + √2 ] [ 3 + 2√2 ]
3 + 5√2 + 4
7 + 5√2