Question 403368
{{{w=c*r^(-2)}}}
"Solve for r" means we want r by itself on one side of the equation. So we want to "remove" the c in front and the exponent of -2. First we will divide by c which, as you will see, will "move" the c over to the other side of the equation:
{{{(w)/c=(c*r^(-2))/c}}}
{{{w/c=(cross(c)*r^(-2))/cross(c)}}}
{{{w/c= r^(-2)}}}
Now we have an equation that tells us what {{{r^(-2)}}} is. We want an equation that tells us what "r" is. The exponent on "r" is a 1. (Any "invisible" exponent is always a 1.) So we want to find a way to change the exponent from a -2 into a 1. To do this we will combine several ideas:<ul><li>It is OK to raise both sides of an equation to the same power.</li><li>The rule for exponents when raising a power to a power is to multiply the exponents.</li><li>Multiplying reciprocals <i>always</i> results in a 1!</li></ul>
So if we raise both sides of the equation to the reciprocal of -2 power, the exponent on r will turn into a 1! The reciprocal of -2 is -1/2. So we will raise both sides of the equation to the -1/2 power:
{{{(w/c)^(-1/2)= (r^(-2))^(-1/2)}}}
On the right side we get {{{r^1}}} or just r, exactly as we planned:
{{{(w/c)^(-1/2)= r}}}
This may be an acceptable form for the answer. Or we could do some things on the left side:
{{{(w/c)^((-1)*(1/2))= r}}}
{{{((w/c)^(-1))^(1/2)= r}}}
{{{(c/w)^(1/2)= r}}}
{{{sqrt(c/w)= r}}}
{{{sqrt((c/w)(w/w))= r}}}
{{{sqrt(cw/w^2)= r}}}
{{{sqrt(cw)/sqrt(w^2)= r}}}
{{{sqrt(cw)/w= r}}}
This may be the preferred form for the answer.