Two skills live here: tidying up an ordinary square root, and taming an expression that goes on forever.
Simplifying a square root
Pull out the biggest perfect-square factor: square×rest=square×rest. So 48=16×3=43. You can only add or subtract roots once the part under the root is the same.
The trick for “forever” expressions
If an expression repeats endlessly, the part inside is a perfect copy of the whole. So name the whole thing x, then replace the inner copy with x too.
• Continued fraction: x=1+x2 → multiply by x → x2−x−2=0 → solve.
Speed shortcut for n+n+…: write n as two consecutive numbers k(k+1); the answer is the larger one, k+1. So 30+… with 30 = 5×6 gives 6.
∞Infinite Nested Radical√(n+√(n+…)) settles to one value
n = 30
x = √(30 + √(30 + √(30 + …)))
Set the whole thing = x, so x = √(30 + x).
Square both sides: x² = 30 + x → x² − x − 30 = 0.
Solve: x = (1 + √(1 + 4×30)) / 2 = 6.000.
Watch it build up: 5.48 → 5.96 → 6.00 → 6.00 → 6.00 → 6.00 → 6.00
30 = 5×6, so x = 6
For 20+20+…: 20 = 4 × 5, so the answer is the larger factor, .
📝Practice Questions
Q1Find x = √(12 + √(12 + √(12 + …))):
Q2Find x = √(6 + √(6 + √(6 + …))):
Q3Simplify √32 + √8:
📚
Real exam questions — Roots & Infinite Expressions
2 question types · 6 solved examples from real SSC papers
Simplify roots by pulling out perfect squares; crack infinite expressions by naming the whole thing x.
How to solve this type
A square root simplifies only when the number inside hides a perfect-square factor: write it as that square times the rest, then square×rest=square×rest. Two roots can be added or subtracted ONLY when the part left under the root (the radicand) is identical — then you just add or subtract the numbers in front. For tricky forms like A+B+A−B, square the whole thing first: the awkward B parts cancel and you are left with a plain number. Never add the numbers inside the roots — a+b is NOT a+b.
