Procedure

Denest $a \pm 2 b$

Algebra • Surds • Nested Radicals

Statement

To simplify

a \pm 2 b

, find

p, q > 0

with

p + q = a

and

pq = b

; then

a \pm 2 b = (p \pm q)^{2}

, so

a \pm 2 b = p \pm q

, taking

p > q

so the value is non-negative.

How to use it

1Force the middle term into the shape $2 b$ first: $43 = 212$ .
2Then look for two numbers with sum $a$ and product $b$ , exactly as when factorising a quadratic.
3If no such rational $p, q$ exist the radical does not denest, and another method is needed.

Simplify

11 + 62

Step-by-step

1
$62 = 218$ , so $a = 11$ , $b = 18$
Why: Put the middle term in the form $2 b$ .
2
$p + q = 11$ , $pq = 18 \Rightarrow p = 9, q = 2$
3
$11 + 62 = (3 + 2)^{2}$
4
$11 + 62 = 3 + 2$
Why: Both parts are positive, so this is the principal root.

Answer

3 + 2

Watch out

With the minus sign the value is

p - q

with

p > q

; writing

q - p

gives a negative number, which a radical never returns.

Builds on

Also known as

nested surd simplification, square root of a plus 2 root b, un-nesting a radical