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
All original content copyright, © 2011-2018. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. Please contact the author if you have questions.
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
All original content copyright, © 2011-2018. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. Please contact the author if you have questions.