Simplifying a Nested Radical (Surd)

The Basic Idea

The basic idea is to simplify an expression of the form sqrt(c+sqrt(d)) such as

sqrt(9/32+sqrt(45)/32)

Let x = sqrt(c+sqrt(d))

Perhaps c+sqrt(d) is a perfect square, in which case this can be simplified.

If it's a perfect square, then it has to be the square of an expression of the form sqrt(a)+sqrt(b).

So c+sqrt(d) = a+b + 2sqrt(ab)

That means c = a+b, and d=4ab

Applying the Idea

Let x = sqrt(9/32+sqrt(45)/32) = sqrt(9/32+sqrt(45/1024))

Here, x has the form sqrt(c+sqrt(d)), where c=9/32 and d=45/1024.

From the "basic idea", above, we see that x=sqrt(a)+sqrt(b) where c=a+b and d=4ab, so a+b=9/32, and 4ab=45/1024.

If you've solved a lot of quadratic equations, then you'll find it easy to find a and b as the roots of y²-(9/32)y+(1/4)(45/1024)=0

However you solve for a and b, you'll see a=3/64, and b=15/64.

So sqrt(9/32+sqrt(45)/32) = sqrt(3/64)+sqrt(15/64) = sqrt(3)/8+sqrt(15)/8

Another example

Let x = sqrt(6-4sqrt(2)) = sqrt(6-sqrt(32))

This is the sqrt of a "perfect square" of the difference of sqrt(a)-sqrt(b), which is solved more or less the same way.

So a+b=6, and 4ab=32

Solving for a and b, a=2 and b=4, so sqrt(6-4sqrt(2)) = sqrt(2)-sqrt(4) = sqrt(2)-2

Oops! This isn't the right answer; the sign is flipped.  This happens, because when we take the square root of both sides, we need to consider both square roots, and throw out the wrong one.  So the answer is 2-sqrt(2).

The Complete Solution for Simplifying sqrt(c+sqrt(d))

Let

sqrt(c+sqrt(d)) = sqrt(a)+sqrt(b).

Square both sides:

c + sqrt(d) = a + b + 2 sqrt(ab)

Equate the rational parts and the irrational parts:

c = a + b
d = 4ab

These equations have two solutions for a and b, so WLOG, I'll let a > b.

a = c/2+sqrt(c²-d)/2
b = c/2-sqrt(c²-d)/2

Therefore,

sqrt(c+sqrt(d)) = sqrt(c/2+sqrt(c²-d)/2) + sqrt(c/2-sqrt(c²-d)/2)

If c²-d is a perfect square, then the RHS of the equation, above, is simpler than the LHS, because it eliminates nested radicals.

The Complete Solution for Simplifying sqrt(c-sqrt(d))

Let

sqrt(c-sqrt(d)) = sqrt(a)-sqrt(b).

Square both sides:

c - sqrt(d) = a + b - 2 sqrt(ab)

Equating the rational parts and the irrational parts gives the same pair of equations, so the final solution is very similar:

sqrt(c-sqrt(d)) = sqrt(c/2+sqrt(c²-d)/2) - sqrt(c/2-sqrt(c²-d)/2)

Again, if c²-d is a perfect square, then the RHS of the equation, above, is simpler than the LHS, because it eliminates nested radicals.

Internet references

IBM's Ponder This Puzzle: September, 2000 - simplify sqrt(3-sqrt5)+sqrt(4+sqrt7)+sqrt(6-sqrt35)

Related pages in this website

Sines and cosines of Special Angles often benefit from this method of simplifying nested radicals. 

Factoring Cubic, Continued -- It's a curious fact that expressions involving nested square and cube roots, which come up in connection with the cubic formula, can be simplified by factoring the cubic equation in question.  For example,

_________ ³√10 + Ö 108  -

__________ ³Ö -10 + Ö 108

can be simplified to an integer: 2.  This page does a thorough analysis of expressions of this type that simplify to integers.

Recurrence Relation of Successive Powers of Polynomial Root -- what a mouthful!  The gist of this topic is that the successive powers of a root of a polynomial form a sequence that has an simple recurrence relation.

The webmaster and author of this Math Help site is Graeme McRae.