fractional powers of x

Question

I would like a step by step explanation of the simplifications made to reach the result:

$$\frac{30x^\frac{1}{2}y}{20x^{\frac{3}{2}}}=\frac{3y}{2x}$$

Answer 1

L.H.S

$$\frac{30yx^\frac{1}{2}}{20x^{\frac{3}{2}}}$$

Simplify:

$$(\frac{3y}{2}) (\frac{x^\frac{1}{2}}{x^\frac{3}{2}})\tag1$$

Apply this power rule: $$\frac{x^a}{x^b}=x^{a-b}$$

(1) becomes: $$(\frac{3y}{2})(x^{\frac{1}{2}-\frac{3}{2}}) \tag2$$

$$x^{\frac{1}{2}-\frac{3}{2}}=x^{\frac{-2}{2}}=x^{-1}=\frac{1}{x}$$

(2) becomes:

$$\frac{3y}{2}\frac{1}{x} $$

Rearranging, the L.H.S:

$$\frac{3y}{2x}$$

Answer 2

There are two things here. First, notice that 30 and 20 share a common factor of 10 so that can be divided out. Second, with the rules of exponents, $\frac{x^m}{x^n} = x^{m-n}$. Now use this to divide $x^{1/2} / x^{3/2}$.

Answer 3

Steps (for future reference):

• $0.$ Come up with a name for whatever difficulty you're having. That will get you closer to the solution to your problem. You already got one, namely, the title of your post.

• $1.$ Google "Properties of (the name of your problem here)". In this case google "Properties of fractional powers" or "Properties of rational exponents". The important word is "properties". I found the following resource: https://www.classzone.com/eservices/home/pdf/student/LA207BAD.pdf It lists six properties.

• $2.$ Try to find the properties that might help you in solving your problem. From the given list, the properties $4$ and $5$ look helpful.

• $3.$ Apply the properties you think are helpful to your problem.