Solver Square Root Arithmetic (Adding, Subtracting, Multiplying, and Dividing Square Roots)

Source code of 'Square Root Arithmetic (Adding, Subtracting, Multiplying, and Dividing Square Roots)'

This Solver (Square Root Arithmetic (Adding, Subtracting, Multiplying, and Dividing Square Roots)) was created by by jim_thompson5910(35256)

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==section input

This solver will show you how to add, subtract, multiply, and divide square roots.

Enter the radicands of the square roots and the operation needed to be done. For instance if you want to add {{{sqrt(2)}}} to {{{sqrt(18)}}}, then enter "2" and "18" into the boxes choose the "+".

Enter positive WHOLE numbers only (ie no fractions or decimal numbers) that will go inside the square root. Also, don't worry about entering "square root", "sqrt", etc...

&#8730;*[input radicand1=2]*[choice operator + - * /]&#8730;*[input radicand2=18]

Please let me know if you find any errors.

==section solution

perl

## --------------------------- Subroutines ------------------------------------------

# given a number n, subroutine will generate all perfect squares (excluding 1) up to n
# eg n=103 ----> perfect squares: 4,9,16,25,36,49,64,81,100
sub generate_perfect_squares
{
 my $n=$_[0];
 my $sqrt_n=int(sqrt($n));
 my @ans;
 for(my $i=2;$i<=$sqrt_n;$i++)
 {push(@ans,$i**2);}
 return @ans;
}

# simplify sqrt(72)=sqrt(36)*sqrt(2)=6*sqrt(2)
sub simplify_square_root
{
 die "Needs one argument" if @_!=1;
 my $number=$_[0];
  
 ## if perfect square, then take square root
 my $square_root=sqrt($number);
 if($square_root!~m/\./)
  {return $square_root;}

my @perfect_squares=generate_perfect_squares($number);
 if(@perfect_squares==0)
 { 
    return "sqrt($number)";
 }

my @factor;

for(my $i=0;$i<@perfect_squares;$i++)
 {
 my $temp=$number/$perfect_squares[$i];
 push(@factor,$perfect_squares[$i]) if $temp!~m/\./;
 }
 if(@factor==0)
 {
 return ("1","sqrt($number)") if ( wantarray );
 return "sqrt($number)";
 }
 my $factor1=$number/$factor[-1];
 my $factor2=$number/$factor1;
 my $sq_factor1=sqrt($factor1);
 my $sq_factor2=sqrt($factor2);
 if($sq_factor1!~m/\./)
 {$factor1=$sq_factor1;}
 else
 {$factor1="sqrt($factor1)";}
 if($sq_factor2!~m/\./)
 {$factor2=$sq_factor2;}
 else
 {$factor2="sqrt($factor2)";}

## Last few lines allow for overloading return values
 #-- undefined context -- return nothing
 return if ( ! defined wantarray );
 #-- list/array context -- return array
 return ($factor2,$factor1) if ( wantarray );
 #-- neither undefined nor list context, return scalar
 return "$factor2*$factor1";
   
}

sub factor_generate
{
 my $number=$_[0];

my ($count,@factors);
 $count=0;
 for(my $i=2;$i<int(sqrt($number))+1;$i++)
 {
 if($number%$i==0)
 {
 $count++;
 push(@factors,$i);
 }
 }
 if($count==0)
 {return "prime";}
 else
 {return @factors;}
}

sub gcf {
  my ($x, $y) = @_;
  ($x, $y) = ($y, $x % $y) while $y;
  return $x;
}

sub lcm {
  return($_[0] * $_[1] / gcf($_[0], $_[1]));
}

# reduce fraction of integer numerator/denominator
sub reduce{
 die "Error: no value entered " if @_==0;
 my @a=generate_fraction($_[0]) if @_==1; ## if input arg is a string
 @a=@_ if @_>1; ## if input arg is an array

if(($a[0]=~m/\./) or ($a[1]=~m/\./))
  {
     return $a[0]/$a[1];
  }

if(@a>2)
  {die "ERROR: More than 2 input arguments";}

foreach (@a)
   {die "Cannot handle decimal numbers " if $_=~m/\./;}

if($a[1]==0)
   {die "ERROR: Division by zero";}

if($a[0]==0)
  {return $a[0];}

my $gcf=gcf($a[0],$a[1]);
 $a[0]/=$gcf;
 $a[1]/=$gcf;
 my $answer_string=join("/",@a);
 $answer_string=~s/\/1$//; # remove denominator of 1

## Last few lines allow for overloading return values
  #-- undefined context -- return nothing
  return if ( ! defined wantarray );
  #-- list/array context -- return array
  return @a if ( wantarray );
  #-- neither undefined nor list context, return scalar
  return $answer_string;
}

#### Square root subroutines

## get EVERY expression in the form of a*sqrt(b)
sub canonical_form
{
   die "Needs 1 argument" if @_!=1;
   my ($exp)=shift;

# if expression doesn't have a square root, add "*sqrt(1)" to end
   $exp.="*sqrt(1)" if($exp!~m/sqrt/);

# if square root doesn't have coefficient (and positive), add "1*" to front
   $exp=~s/^sqrt/1*sqrt/;

# if square root doesn't have coefficient (and negative), add "-1*" to front
   $exp=~s/^\-sqrt/-1*sqrt/;

#make sure that there is a * in between the coefficient and the "sqrt"
   $exp=~s/(\d)sqrt/$1\*sqrt/;

# remove any duplicate asterisks
   $exp=~s/\*+/\*/g;

return $exp;
}

## pull out coefficient "a" and radicand "b" from a*sqrt(b)
sub extract_coefficient_radicand
{
   die "Needs 1 argument" if @_!=1;
   my $exp=shift;

## get into canonical and simplify square root
   $exp=canonical_form($exp);

# extract coefficient and radicand
   my ($coefficient,$radicand)=$exp=~m/(.*?)\*sqrt$(.*?)$/;

return ($coefficient,$radicand);
}

## ------------------------------ End Subroutines --------------------------------------------

if(($radicand1=~m/\D/)||($radicand2=~m/\D/))
{
 print " Error: Please enter POSITIVE and WHOLE numbers only ";
 return; 
}

my $simplify_square_root_note="(note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)";

## Adding two square roots.
if($operator eq '+')
{
   my $ans;

print " {{{sqrt($radicand1)+sqrt($radicand2)}}} Start with the given expression. ";
 my $simplified_root1=simplify_square_root($radicand1);
 $simplified_root1=~s/^1\*sqrt/sqrt/;
 my $simplified_root2=simplify_square_root($radicand2);
 $simplified_root2=~s/^1\*sqrt/sqrt/;

if($simplified_root1 ne "sqrt($radicand1)")
 {
 print " {{{$simplified_root1+sqrt($radicand2)}}} Simplify the first square root {{{sqrt($radicand1)}}} to get {{{$simplified_root1}}}";

print "$simplify_square_root_note ";
 }

if($simplified_root2 ne "sqrt($radicand2)")
 {
 print " {{{$simplified_root1+$simplified_root2}}} Simplify the second square root {{{sqrt($radicand2)}}} to get {{{$simplified_root2}}}";

print "$simplify_square_root_note " if $simplified_root1 eq "sqrt($radicand1)";

print " " if $simplified_root1 ne "sqrt($radicand1)";

}

my ($simplified_coefficient1,$simplified_radicand1);   
   my ($simplified_coefficient2,$simplified_radicand2);

($simplified_coefficient1,$simplified_radicand1)=extract_coefficient_radicand($simplified_root1);
  ($simplified_coefficient2,$simplified_radicand2)=extract_coefficient_radicand($simplified_root2);

if(($simplified_root1 eq "sqrt($radicand1)") and ($simplified_root2 eq "sqrt($radicand2)"))
 {
 if($simplified_radicand1 ne $simplified_radicand2)
 {
 print " Since we have no common terms, this means that we cannot add {{{sqrt($radicand1)}}} and {{{sqrt($radicand2)}}} ";
 return;
 } 
 }

my $final_coefficient=$simplified_coefficient1+$simplified_coefficient2;

## if radicands are equal, combine like terms
 if($simplified_radicand1 eq $simplified_radicand2)
 {
 if($simplified_radicand1 ne '1')
 {
 print " Take note that we have the common term {{{sqrt($simplified_radicand1)}}} ";
 print " {{{($simplified_coefficient1+$simplified_coefficient2)sqrt($simplified_radicand1)}}} Factor out the common term. ";

print " {{{$final_coefficient*sqrt($simplified_radicand1)}}} Combine like terms. ";
 $ans="$final_coefficient*sqrt($simplified_radicand1)";
 }
 else
 {
 print " {{{$final_coefficient}}} Combine like terms. ";
 $ans=$final_coefficient;
 }

print " So {{{sqrt($radicand1)+sqrt($radicand2)=$ans}}} ";
 
 
 }
 else
 {
 print " Since we do NOT have any common terms, this means that we must stop here. So {{{sqrt($radicand1)+sqrt($radicand2)=$simplified_root1+$simplified_root2}}} ";

}

## Subtracting two square roots.
if($operator eq '-')
{
   my $ans;

print " {{{sqrt($radicand1)-sqrt($radicand2)}}} Start with the given expression. ";
 my $simplified_root1=simplify_square_root($radicand1);
 $simplified_root1=~s/^1\*sqrt/sqrt/;
 my $simplified_root2=simplify_square_root($radicand2);
 $simplified_root2=~s/^1\*sqrt/sqrt/;

if($simplified_root1 ne "sqrt($radicand1)")
 {
 print " {{{$simplified_root1-sqrt($radicand2)}}} Simplify the first square root {{{sqrt($radicand1)}}} to get {{{$simplified_root1}}}";

print "$simplify_square_root_note ";
 }

if($simplified_root2 ne "sqrt($radicand2)")
 {
 print " {{{$simplified_root1-$simplified_root2}}} Simplify the second square root {{{sqrt($radicand2)}}} to get {{{$simplified_root2}}}";

print "$simplify_square_root_note " if $simplified_root1 eq "sqrt($radicand1)";

print " " if $simplified_root1 ne "sqrt($radicand1)";

}

my ($simplified_coefficient1,$simplified_radicand1);   
   my ($simplified_coefficient2,$simplified_radicand2);

($simplified_coefficient1,$simplified_radicand1)=extract_coefficient_radicand($simplified_root1);
  ($simplified_coefficient2,$simplified_radicand2)=extract_coefficient_radicand($simplified_root2);

if(($simplified_root1 eq "sqrt($radicand1)") and ($simplified_root2 eq "sqrt($radicand2)"))
 {
 if($simplified_radicand1 ne $simplified_radicand2)
 {
 print " Since we have no common terms, this means that we cannot subtract {{{sqrt($radicand1)}}} and {{{sqrt($radicand2)}}} ";
 return;
 } 
 }

my $final_coefficient=$simplified_coefficient1-$simplified_coefficient2;

## if radicands are equal, combine like terms
 if($simplified_radicand1 eq $simplified_radicand2)
 {
 if($simplified_radicand1 ne '1')
 {
 print " Take note that we have the common term {{{sqrt($simplified_radicand1)}}} ";
 print " {{{($simplified_coefficient1-$simplified_coefficient2)sqrt($simplified_radicand1)}}} Factor out the common term. ";

print " So {{{sqrt($radicand1)-sqrt($radicand2)=$ans}}} ";
 
 
 }
 else
 {
 print " Since we do NOT have any common terms, this means that we must stop here. So {{{sqrt($radicand1)-sqrt($radicand2)=$simplified_root1-$simplified_root2}}} ";

}

## Multiplying two square roots.
if($operator eq '*')
{
   my $ans;

if($radicand1 eq 0)
 {
 print " Multiplying 0 by ANY number will give you zero. In other words, {{{0*x=0}}}. ";

print " This means that {{{sqrt($radicand1)*sqrt($radicand2)=0*sqrt($radicand2)=0}}}. ";

print " So {{{sqrt($radicand1)*sqrt($radicand2)=0}}}. ";
 return;
 }

if($radicand2 eq 0)
 {
 print " Multiplying 0 by ANY number will give you zero. In other words, {{{x*0=0}}}. ";

print " This means that {{{sqrt($radicand1)*sqrt($radicand2)=sqrt($radicand1)*0=0}}}. ";

print " So {{{sqrt($radicand1)*sqrt($radicand2)=0}}}. ";
 return;
 }

print " {{{sqrt($radicand1)*sqrt($radicand2)}}} Start with the given expression. ";
 
 print " {{{sqrt($radicand1*$radicand2)}}} Combine the square roots. Remember, {{{sqrt(x)*sqrt(y)=sqrt(x*y)}}} ";

my $new_radicand=$radicand1*$radicand2;

print " {{{sqrt($new_radicand)}}} Multiply. ";

my $simplified_root=simplify_square_root($new_radicand);
   $simplified_root=~s/^1\*sqrt/sqrt/;

if($simplified_root ne "sqrt($new_radicand)")
 {
 print " {{{$simplified_root}}} Simplify the square root. $simplify_square_root_note ";
 $ans=$simplified_root;
 }
 else
 {
 $ans="sqrt($new_radicand)";
 }

print " So {{{sqrt($radicand1)*sqrt($radicand2)=$ans}}} ";

}

## Multiplying two square roots.
if($operator eq '/')
{
   my $ans;

if($radicand1 eq 0)
 {
 print " Dividing 0 by ANY number will give you zero. In other words, {{{0/x=0}}}. ";

print " This means that {{{sqrt($radicand1)/sqrt($radicand2)=0/sqrt($radicand2)=0}}}. ";

print " So {{{sqrt($radicand1)/sqrt($radicand2)=0}}}. ";
 return;
 }

if($radicand2 eq 0)
 {
 print " Error: Division by zero. You CANNOT divide by zero. ";
 return; 
 }

print " {{{sqrt($radicand1)/sqrt($radicand2)}}} Start with the given expression. ";

my $temp_sq=sqrt($radicand2);

## see if denominator is perfect square
 if($temp_sq!~/\./)
 {
 print " {{{sqrt($radicand1)/$temp_sq}}} Take the square root of the denominator ";

my $simplified_num=simplify_square_root($radicand1);
      $simplified_num=~s/^1\*sqrt/sqrt/;

if($simplified_num ne "sqrt($radicand1)")
 {
 print " {{{($simplified_num)/$temp_sq}}} Simplify the numerator. $simplify_square_root_note ";
 $ans=$simplified_root;

my ($coefficient,$temp_rad)=extract_coefficient_radicand($simplified_num);

my ($temp_num,$temp_denom)=reduce($coefficient,$temp_sq);

if("$temp_num/$temp_denom" ne "$coefficient/$temp_sq")
 {
 $ans="($temp_num*sqrt($temp_rad))/$temp_denom";
 $ans=~s/^$1\*/\(/;
 $ans=~s/^\(sqrt\((\d+)$\)/sqrt$$1$/;
 $ans=~s/sqrt$1$/1/;
 $ans=~s/$(\d+)\*1$/$1/;
 print " {{{$ans}}} Reduce. ";
 }
 else
 {
 $ans="($temp_num*sqrt($temp_rad))/$temp_denom";
 }

}
 else
 {
 $ans="sqrt($radicand1)/$temp_sq";
 }
 }
 else
 {
 print " {{{(sqrt($radicand1)*sqrt($radicand2))/(sqrt($radicand2)*sqrt($radicand2))}}} Multiply both numerator and denominator by {{{sqrt($radicand2)}}}. ";

print " {{{(sqrt($radicand1*$radicand2))/(sqrt($radicand2*$radicand2))}}} Combine the radicals. Remember, {{{sqrt(x)*sqrt(y)=sqrt(x*y)}}}. ";

my $new_num_radicand=$radicand1*$radicand2;
      my $new_denom_radicand=$radicand2*$radicand2;

print " {{{(sqrt($new_num_radicand))/(sqrt($new_denom_radicand))}}} Multiply. ";

print " {{{(sqrt($new_num_radicand))/($radicand2)}}} Take the square root of {{{$new_denom_radicand}}} to get {{{$radicand2}}}. Notice how {{{sqrt($new_denom_radicand)=sqrt($radicand2^2)=$radicand2}}} ";

my $simplified_num=simplify_square_root($new_num_radicand);
      $simplified_num=~s/^1\*sqrt/sqrt/;

if($simplified_num ne "sqrt($new_num_radicand)")
 {
 print " {{{($simplified_num)/$radicand2}}} Simplify the numerator. $simplify_square_root_note ";
 $ans=$simplified_root;

my ($coefficient,$temp_rad)=extract_coefficient_radicand($simplified_num);

my ($temp_num,$temp_denom)=reduce($coefficient,$radicand2);

if("$temp_num/$temp_denom" ne "$coefficient/$radicand2")
 {
 $ans="($temp_num*sqrt($temp_rad))/$temp_denom";
 $ans=~s/^$1\*/\(/;
 $ans=~s/^\(sqrt\((\d+)$\)/sqrt$$1$/;
 $ans=~s/sqrt$1$/1/;
 $ans=~s/$(\d+)\*1$/$1/;
 print " {{{$ans}}} Reduce. ";
 }
 else
 {
 $ans="($temp_num*sqrt($temp_rad))/$temp_denom";
 }

}
      else
      {
         $ans="sqrt($new_num_radicand)/$radicand2";
      }
   
   }

print " So {{{sqrt($radicand1)/sqrt($radicand2)=$ans}}} ";

}
   
==section output

==section check