Simple Fraction needing explanation

Question

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

How? I don't see what is going on here. What rule is being used to achieve this amount?

Answer 1

Use $$\frac{x^a}{x^b} = x^{a-b} $$ so $$\frac{x}{x^{-1/2}} =\frac{x^1}{x^{-1/2}}= x^{1-(-1/2)}=x^{3/2}$$

Answer 2

Use: $x^ax^b=x^{a+b}$:

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

Answer 3

$$\frac{x^a}{x^b} = x^{a-b} $$

Answer 4

By definition of negative exponent, $\displaystyle x^{-a}=\frac{1}{x^a}$. As a consequence, $\displaystyle x^{a}=\frac{1}{x^{-a}}$. Thus,

$$\frac{x}{x^{-1/2}}=x\cdot\frac{1}{x^{-1/2}}=x\cdot x^{1/2}$$

Using the"product of powers rule", we get the desired result.