Find the value of √(388 + √(127 + √289))

Correct — A. Clear the roots from the inside out.

Innermost root: √289 = 17
Add: 127 + 17 = 144
Middle root: √144 = 12

Add: 388 + 12 = 400
Outer root: √400 = 20 → option (a).

Why the others are wrong

• (b)
  30 — 30² = 900, so 30 would need 900 under the outer root. The working puts 388 + 12 = 400 there, and √400 = 20.
• (c)
  40 — 40² = 1600, four times the 400 that actually sits under the outer root. Squaring each option and comparing with 400 rules 40 out in one step.
• (d)
  50 — 50² = 2500, over six times the 400 under the outer root. 388 plus the small middle root (12) cannot come anywhere near 2500.

A nested root built like this collapses layer by layer: each inner result, added to the next number, makes a perfect square.

Here 127 + 17 = 144 = 12² and 388 + 12 = 400 = 20². Knowing the squares up to 30² lets you see each step at once.

A check from the options: square each option. Only 20² = 400 sits just above 388.

Key facts

• √289 = 17, √144 = 12 and √400 = 20.
• Nested roots are evaluated from the innermost root outward, like nested brackets.
• √(a + b) is not √a + √b: √(388 + 12) = 20, while √388 + √12 ≈ 19.70 + 3.46 = 23.16.

Common traps

• Splitting a root across a sum, as if √(388 + 12) were √388 + √12.
• Starting at the outer root and trying to take √388 before the inner roots are cleared.

No directly related past PYQ was found.