TI-84+ and Casio (fx-CG 50) Micropython: Simplifying Nested Radicals

TI-84+ and Casio (fx-CG 50) Micropython: Simplifying Nested Radicals

Introduction 

The following program will simplifying the following expression:

√(x + y) = √a + √b

The input is x and y, with output a and b.  The following conditions are implied:  x > y and x > 0.

To see the derivation of simplification and examples, please check out this blog entry:

https://edspi31415.blogspot.com/2018/11/simplifying-nested-radicals.html

TI 84 Plus Program:  DENEST

"EWS 2018-11-12"
Disp "√(X+Y)=√(A)+√(B)","X>Y","X>0"
Prompt X,Y
X/2+1/2*√(X^2-Y^2)→A
X/2-1/2*√(X^2-Y^2) →B
√(A)→Z
round(Z,9)→Z
Disp " "

If Y<0 font="">

Then
If fPart(Z)=0
Then 
Disp √(A),"- √(",B,")"
Else 
Disp "√(",A,")- √(",B,")"
End

Else
If fPart(Z)=0
Then 
Disp √(A),"+ √(",B,")"
Else 
Disp "√(",A,")+ √(",B,")"
End

End

Casio Micropython (fx-CG 50) Script denest.py

Input is the form of:

√(x1 * √x2 + y1 * √y2) = √a + √b

import math
print("sqrt(x+y)=")
print("sqrt(a)+sqrt(b)")
print("x>y","x>0")

print(" ")
print("x1*sqrt(x2)")
x1=float(input("x1:"))
x2=float(input("x2:"))
x=x1*math.sqrt(x2)

print(" ")
print("y1*sqrt(y2)")
y1=float(input("y1:"))
y2=float(input("y2:"))
y=y1*math.sqrt(y2)

a=x/2+math.sqrt(x**2-y**2)
b=x/2-math.sqrt(x**2-y**2)
z=math.sqrt(a)
z=round(z,10)
a=round(a,10)
b=round(b,10)

if y<0: font="">
  if z-int(z)==0:
    print(z,"-sqrt(")
    print(b,"?")
  else:
    print("sqrt(",a)
    print("-sqrt(")
    print(b,")")
else:
  if z-int(z)==0:
    print(z,"+sqrt(")
    print(b,")")
  else:
    print("sqrt(",a)
    print("+sqrt(")
    print(b,")")


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.

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

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.