SOLUTION: {{{ (x^(3/2)+3x^(-3/2))/g^(1/2) }}}

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Question 249786: +%28x%5E%283%2F2%29%2B3x%5E%28-3%2F2%29%29%2Fg%5E%281%2F2%29+
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+%28x%5E%283%2F2%29%2B3x%5E%28-3%2F2%29%29%2Fg%5E%281%2F2%29+
I assume the problem is to simplify this expression.

As I hope you know, fractional exponents mean a root of some kind. All of these exponents have a denominator of 2 so these are all square roots. Simplified fractions do not have square roots in denominators (even if they are disguised with fractional exponents).

This expression has two square roots in denominators. g%5E%281%2F2%29 is obviously in the denominator. The other square root in a denominator is not as easy to see. It is the x%5E%28-3%2F2%29. Negative exponents represent reciprocals, as I hope you know. So x%5E%28-3%2F2%29+=+1%2Fx%5E%283%2F2%29 and now we can see the square root in the denominator.

Another part of simplifying fractions is to eliminate any fractions within a fraction. As we saw just above, x%5E%28-3%2F2%29 which is part of the numerator represents a fraction. So we need to address this issue, too.

To simplify we are going to eliminate these two square roots in denominators and the fraction within a fraction by using basic Algebra and the rules for exponents. Part of the basic Algebra is that we are allowed to change fractions by multiplying the numerator and denominator by the same thing. After all a fraction with the same numerator and denominator is simply a 1 and multiplying by 1 never really changes an expression. (It will look different but it will be the same expression in essence.)

We can eliminate the one of these square roots by multiplying the numerator and denominator by g%5E%281%2F2%29:

When we multiply this, we will use the rule for exponents when we multiply the g's in the denominators:

which simplifies to:
+%28g%5E%281%2F2%29%2Ax%5E%283%2F2%29%2B3g%5E%281%2F2%29%2Ax%5E%28-3%2F2%29%29%2Fg%5E1
or
+%28g%5E%281%2F2%29%2Ax%5E%283%2F2%29%2B3g%5E%281%2F2%29%2Ax%5E%28-3%2F2%29%29%2Fg
and we have eliminated one of the square roots in a denominator.

Now we need to deal with the x%5E%28-3%2F2%29. As mentioned earlier, there are two issues here: a fraction within a fraction and a square root in a denominator. To eliminate the fraction within a fraction, we want to change the exponent of x%5E%28-3%2F2%29 so that it is not negative. The simplest way to do this is to multiply the numerator and denominator by x%5E%283%2F2%29:


Using the rule for exponents when we multiply the x's:

which simplifies to:
+%28g%5E%281%2F2%29%2Ax%5E%286%2F2%29%2B3g%5E%281%2F2%29%2Ax%5E0%29%2Fgx%5E%283%2F2%29
Since x%5E0+=+1:
+%28g%5E%281%2F2%29%2Ax%5E%286%2F2%29%2B3g%5E%281%2F2%29%29%2Fgx%5E%283%2F2%29
The fraction within a fraction is now gone. But we still have a square root in the denominator: x%5E%283%2F2%29. So we can eliminate it by multiplying by x%5E%281%2F2%29:




+%28g%5E%281%2F2%29%2Ax%5E%287%2F2%29%2B3g%5E%281%2F2%29%2Ax%5E%281%2F2%29%29%2Fgx%5E2
The square roots in denominators are gone and the fraction within a fraction is gone. This is a simplified expression.

P.S. If we were really clever we could have done all this in one big step: Multiply the original fraction by %28g%5E%281%2F2%29x%5E2%2Fg%5E%281%2F2%29x%5E2%29. You might want to try this yourself.