Question

\sqrt{7+\sqrt{2+\sqrt{x+1} } } =3

Answer 1

First make note of the domain of the left hand side.

• √(x + 1) is defined for x + 1 ≥ 0, or x ≥ -1.

• √(2 + √(x + 1)) is defined for 2 + √(x + 1) ≥ 0. If x ≥ -1, then this condition is satisfied.

• √(7 + √(2 + √(x + 1))) is defined for 7 + √(2 + √(x + 1)) ≥ 0. This condition is also satisified automatically if x ≥ -1.

Now solve the equation. Taking the square of both sides of

√(7 + √(2 + √(x + 1))) = 3

yields

7 + √(2 + √(x + 1)) = 3² = 9

√(2 + √(x + 1)) = 2

Squaring both sides again gives

2 + √(x + 1) = 2² = 4

√(x + 1) = 2

And once more,

x + 1 = 2² = 4   ⇒   x = 3