Solver Simplifying Square Roots (whole numbers only)

Source code of 'Simplifying Square Roots (whole numbers only)'

This Solver (Simplifying Square Roots (whole numbers only)) was created by by jim_thompson5910(28595)

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==section input
Enter whole numbers only for the radicand. For instance, if you want to simplify {{{sqrt(75)}}}, enter 75 into the box:

*[input input=75]

==section solution

perl

if(($input=~m/\D/)&&($input!~m/\-/))
{print "<font color=red>Error: Enter whole numbers only</font>";
return undef;}

#if($input<0)
#{print "<font color=red>Error: Enter positive values only</font>
#
#Simply factor out a negative 1 to make the value positive. 
#
#So change {{{sqrt($input)}}} to {{{i*sqrt(",abs($input),")}}} where {{{i=sqrt(-1)}}}";
#return undef;}

if($input<0)
    {
    $in_negative_check=1;

print "

{{{sqrt($input)}}} Start with the given expression ";
    $input=abs($input);
    print "

{{{sqrt(-1*$input)}}} Factor out a negative 1

{{{sqrt(-1)*sqrt($input)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}

{{{i*sqrt($input)}}} Replace {{{sqrt(-1)}}} with {{{i}}} (remember {{{i=sqrt(-1)}}})

Now lets simplify {{{sqrt($input)}}}:

";}

$sq=sqrt($input);

$value=$input;

$input="sqrt($input)";

$square_root_value=sqrt($value);

#$input="sqrt(75)";

if($input=~ m/(\d+)/)
{$num=$1;}

if($in_negative_check!=1)
{print "{{{$input}}} Start with the given expression";}

if($sq!~/\./)
{print "

Since $value is a perfect square, we can take the square root of $value to get a clean number

{{{$input=$sq}}}";

if($in_negative_check==1)
{print "

So {{{sqrt(-$value)}}} simplifies to {{{$sq*i}}} (just reintroduce {{{i}}} back in)";}

return;}

$tick=0;

for($count=2;$count<$num;$count++)
{
 if($num%$count==0)
  {$gcf[$tick]=$count;
  $tick++;}
}

print "

The goal of simplifying expressions with square roots is to factor the radicand into a product of two numbers. One of these two numbers must be a perfect square.  When you take the square root of this perfect square, you will get a rational number.";

if($tick==0)
{print "

However, since $num is prime, we cannot factor $num any further. So the expression {{{$input}}} cannot be reduced any further";

if($in_negative_check==1)
{print "

So {{{sqrt(-$value)}}} can only be simplified to {{{i*sqrt($value)}}}";}

return;}

print "

So let's list the factors of $num";

print "

Factors:

1,";

for($count=0;$count<($size=@gcf);$count++)
{
 $sq=sqrt($gcf[$count]);
 print " $gcf[$count],";
 if($sq!~/\./)
  {#print "\n $gcf[$count] is a perfect square \n";
  $num1=$gcf[$count]}
}

if($num1==0)
{print "$num

Since none of these factors are perfect squares, we cannot simplify {{{$input}}}

if($in_negative_check==1)
{print "

So {{{sqrt(-$value)}}} can only be simplified to {{{i*sqrt($value)}}}";}
}

if($num1!=0)
{$num2=$num/$num1;

$sq1=sqrt($num2);

print " $num

Notice how $num1 is the largest perfect square, so lets factor $num into $num1*$num2

print "

{{{sqrt($num1*$num2)}}} Factor $num into $num1*$num2

print "

{{{sqrt($num1)*sqrt($num2)}}} Break up the square roots using the identity {{{sqrt(x*y)=sqrt(x)*sqrt(y)}}}

$sq=sqrt($num1);

print "

{{{$sq*sqrt($num2)}}} Take the square root of the perfect square $num1 to get $sq

if($sq1!~/\./)
{

$temp=$sq*$sq1;
print "{{{$sq*$sq1}}} Take the square root of the perfect square $num2 to get $sq1

{{{$temp}}} Multiply

print "
So the expression {{{$input}}} simplifies to {{{$temp}}}";}

if($sq1=~/\./)
{$temp="$sq*sqrt($num2)";
print "

So the expression {{{$input}}} simplifies to {{{$sq*sqrt($num2)}}}

";}

if($in_negative_check==1)
{$temp=~s/(\d+)\*sqrt/\1\*i\*sqrt/;

print "

---------------------
Answer:

So the expression

{{{sqrt(-$value)}}}

simplifies to

{{{$temp}}} (just reintroduce {{{i}}} back in)

return;}

print "

----------------------------
Check:

Notice if we evaluate the square root of $value with a calculator we get

{{{$input=$square_root_value}}}

and if we evaluate {{{$temp}}} we get

{{{$temp=$square_root_value}}}

This shows that {{{$input=$temp}}}. So this verifies our answer  ";

}

==section output

==section check
  angle=40 angle1=50