Question 980262
I may know where you are because I have tutored meek high school girls with
pushy engineering student older brothers who kept making them fell dumb.
I think I can help.
Simplifying square roots has very little to do with the mass and volume of the sun,
which should be related to the cube (not the square) of its radius.
A calculator can help you simplify square roots (and cube roots),
but you mostly have to use your own head,
and quick recall of those pesky multiplication facts
that teachers and parents kept insisting that you should memorize,
but they seemed so useless.
 
Simplifying square roots means going from {{{sqrt(12)}}} to {{{2sqrt(3)}}} ,
and things like that.
To do that you need to know that
{{{12=4*3}}} and that {{{sqrt(12)=sqrt(4*3)=sqrt(4)*sqrt(3)=2*sqrt(3)}}} .
The hard part of that is finding the factors that will be useful.
Knowing that {{{12=6*2}}} does not help;
you need to find that {{{4=2^2}}} factor that is a perfect square.
 
EXAMPLE:
Say you have to simplify {{{sqrt(78408)}}} .
There may be a perfect square that is a factor of {{{78408}}} ,
with {{{78408=square*something}}} ,
but how do you find that {{{square}}} factor?
That square factor may be the square of a simple prime number,
like 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.
It may also be the square of a number made by multiplying those prime numbers,
like {{{2*3=6}}} , or {{{2*7=14}}} , or {{{3*5=15}}} , or {{{3*11*17=561}}} ,
or any other combination.
Knowledge that is already in your head may help you find factors of {{{78408}}} ,
but the calculator can help too, if you know how to use it.
I look at {{{78408}}} and I can see that {{{4=2^2}}} and {{{9=3^2}}} are factors,
but the calculator can tell you the same thing, and more:
{{{78408/4=19602}}}<--->{{{78408=4*19602}}} and
{{{19602/9=2178}}}<--->{{{19602=9*2178}}} .
So, {{{78408=4*19602=4*9*2178=36*2178}}} .
Can we get any other perfect square factor out of {{{2178}}} ?
The calculator tells me that {{{2178/4=544.5}}} , which is not an integer,
so {{{2178}}} does not divide evenly by {{{4}}} ,
meaning that {{{4}}} is not a factor of {{{2178}}} .
However, {{{2178/9=242}}}<--->{{{2178=9*242}}} .
So far we have that
{{{78408=4*19602=4*9*2178=4*9*9*242}}} .
Now we want to find factor of {{{242}}} ,
perfect square factors if possible.
If you realize that {{{242=2*121}}} and {{{121=11^2}}} ,
you are done (or almost done).
You have pulled out of {{{78408}}} all the perfect square factors,
and were left with {{{2}}} , which is not a perfect square.
Otherwise, you keep trying dividing by the squares of prime numbers,
until you find that {{{242/121=2}} .
Anyway, you finally find that
{{{78408=4*19602=4*9*2178=4*9*9*242=4*9*9*121*2}}} ,
and the only factor listed that is not a perfect square is {{{2}}} .
So,
{{{sqrt(78408)=sqrt(4*9*9*121*2)=sqrt(4)*sqrt(9)*sqrt(9)*sqrt(121)*sqrt(2)=
2*3*3*11*sqrt(2)=198sqrt(2)}}} .
 
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