3 Ways of Evaluating Nested Square Roots

Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to understand how the students can handle “…”. This enables us to setup a quadratic equation to evaluate its exact value using the quadratic formula,
$$x= \frac{-b \ \pm \sqrt{b^2 – 4ac}}{2a}$$.
Let’s take a look at the following worked examples for finding the nested square roots.

Worked Example 1 of Nested Square Roots

Evaluate \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}} \).

Worked Example 2 of Nested Square Roots

Evaluate \( \sqrt{2 – \sqrt{2 – \sqrt{2 – \sqrt{2 – \ldots}}}} \).

Worked Example 3 of Nested Square Roots

Evaluate \( \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \ldots}}}} \) using the trigonometric property.

Great! Hope these help.


  1. George Plousos

    What would you say about that?
    √2±(√2±(√2±(√2± … ±(√2)…)))=?
    For any combination of signs.

    1. iitutor Post author

      It is a mixture of example 1 and 2. Which means the answer is either 2 or 0.

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