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Simplifying Nested Radicals

Simplifying Nested Radicals

Introduction

A nested radical is a radical of the form:

√(x_0 + √(x_1 + √(x_2 + …. + √(x_k) … )))

A expression of radicals (not necessarily square roots) that contains radicals or surds (unresolved n th roots).  Example of a nested radicals include:

√(8 +  √3)

√(8 +  2 * √3)

√(√8 +  √3)

√(√8 –  √3)

This blog entry will deal with the denesting the following:

√(x + y) and √(x – y) where x and/or y is a radical.  Let’s assume that x > y and only principal roots are calculated. We’re looking into a neat process that is used in computer algebra systems.

Derivation

We want to obtain the following and determine a and b. 

(I) √(x + y) = √a + √b

and

(II) √(x – y) = √a – √b

Start by squaring both sides of (I):

(√(x + y))^2 = (√a + √b)^2

(III) x + y = a + 2 * √(a*b) + b

(IV)  Set:
x = a + b
y = 2 * √(a*b)

(V)  Determine  x – y:

x – y = a + -1*2 * √(a*b) + b  (see IV)
x – y = a – 2 * √(a*b) + b
√(x – y) = √a – √b

(VI)

√(x + y) * √(x – y) = (√a + √b) * (√a – √b)
√((x + y) * (x – y)) = (√a)^2 – (√b)^2
√(x^2 – y^2) = a – b

We have the following simultaneous equations set up:

(VII)
a + b = x     (from (IV))
a – b = √(x^2 – y^2)   (from (VI))

Solving the system in (VII) for both a and b yield:

(VIII)
a = x/2 + 1/2 * √(x^2 – y^2)
b = x/2 – 1/2 * √(x^2 – y^2)

Substituting a and b back in (I) and (II):

(IX)

√(x + y) = √(x/2 + 1/2 * √(x^2 – y^2)) + √(x/2 – 1/2 * √(x^2 – y^2))

√(x – y) = √(x/2 + 1/2 * √(x^2 – y^2)) – √(x/2 – 1/2 * √(x^2 – y^2))

Examples

Example 1:  Simplify  √(8 + 2 * √15)

x = 8
y = 2 * √15

x^2 – y^2
= 8^2 – (2 * √15)^2
= 64 –  4 * 15
= 4

Then: 

√(x/2 + 1/2 * √(x^2 – y^2))
= √(4 + 1/2 * √4)
= √(4 + 1/2 * 2)
= √5    // √a

And:

√(x/2 – 1/2 * √(x^2 – y^2))
= √(4 – 1/2 * √4)
= √(4 – 1/2 * 2)
= √3     // √b

Hence:

√(8 + 2 * √15) = √5 + √3

Example 2:  Simplify  √(88 + 2 * √567)

x = 88
y = 2 * √567

x^2 – y^2
= 88^2 – (2 * √567)^2
= 7744 – 2268
= 5476

Note √5476 = 74

Then: 

√(x/2 + 1/2 * √(x^2 – y^2))
= √(44 + 1/2 * 74)
= √81
= 9        // √a

And:

√(x/2 + 1/2 * √(x^2 – y^2))
= √(44  – 1/2 * 74)
= √7      // √b

√(88 + 2 * √567) = 9 + √7

Example 3:  Simplify √(19 – 4 * √22))  (note the subtraction)

x = 19
y = 4 * √22

x^2 – y^2
= 19^2 – (4 * √22)^2
= 9

√(x/2 + 1/2 * √(x^2 – y^2))  = √11
√(x/2 + 1/2 * √(x^2 – y^2))  = √8 = 2 * √2

Hence:

√(19 – 4 * √22))  = √11 – 2 * √2

Neat algebra.

Source:

Michael J. Wester, Editor.  Computer Algebra Systems: A Practical Guide John Wiley & Sons: Chichester 1999.  ISBN 978-0-471-983538

Eddie

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