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Nested Radical — Definition, Formula & Examples

A nested radical is a radical expression where one radical appears inside another, such as 2+3\sqrt{2 + \sqrt{3}}. Simplifying these expressions often involves rewriting them as a single, simpler radical or as a sum of radicals.

A nested radical is an expression of the form a+b\sqrt{a + \sqrt{b}} (or with subtraction, higher-index roots, or deeper nesting), where the radicand of an outer radical itself contains a radical. Denesting is the process of expressing such a form as a radical-free combination or a simpler radical expression, when one exists.

How It Works

To denest a+b\sqrt{a + \sqrt{b}}, you look for values pp and qq such that a+b=p+q\sqrt{a + \sqrt{b}} = \sqrt{p} + \sqrt{q}. Squaring both sides gives a+b=p+q+2pqa + \sqrt{b} = p + q + 2\sqrt{pq}. Matching rational and irrational parts, you get p+q=ap + q = a and 4pq=b4pq = b. This means pp and qq are roots of t2−at+b4=0t^2 - at + \frac{b}{4} = 0. If the discriminant a2−ba^2 - b is a perfect square, the nested radical can be denested cleanly.

Worked Example

Problem: Simplify 3+22\sqrt{3 + 2\sqrt{2}}.
Set up: Assume the expression equals p+q\sqrt{p} + \sqrt{q}. Squaring both sides gives:
3+22=p+q+2pq3 + 2\sqrt{2} = p + q + 2\sqrt{pq}
Match parts: Matching the rational parts and the irrational parts separately:
p+q=3,pq=2p + q = 3, \quad pq = 2
Solve: The values pp and qq satisfy t2−3t+2=0t^2 - 3t + 2 = 0, which factors as (t−1)(t−2)=0(t-1)(t-2) = 0. So p=2p = 2 and q=1q = 1.
3+22=2+1=2+1\sqrt{3 + 2\sqrt{2}} = \sqrt{2} + \sqrt{1} = \sqrt{2} + 1
Answer: 3+22=1+2\sqrt{3 + 2\sqrt{2}} = 1 + \sqrt{2}

Why It Matters

Nested radicals appear in exact trigonometric values (like cos⁡15°\cos 15°), solutions to quartic equations, and competition mathematics. Being able to denest them lets you work with cleaner expressions and verify algebraic results in precalculus and beyond.

Common Mistakes

Mistake: Trying to denest by "distributing" the outer radical into the sum, writing a+b\sqrt{a + \sqrt{b}} as a+b\sqrt{a} + \sqrt{\sqrt{b}}.
Correction: You cannot split a radical over addition. Instead, use the squaring method: assume the result is p+q\sqrt{p} + \sqrt{q} and solve for pp and qq by matching parts.