How can you express an equation as a quotient if there are exponential fractions and variables involved?

Question

the question is    x^-1/2- x^3/2  and the "objective" is to express as a quotient. is it possible to solve in the exponential fraction form? would the answer be x^2/2 which is just x? Or does each variable with an exponent need to be turned into radical form?

Answer 1

.             1
 x ^{- 1} = ——
.             x
x^1/2 = √x  
~~~~~~~~~~~~~
  x ^{- 1/2}  –   x ^3/2  = 
.   1   
  ——   –  x^3/2  = 
.   x^1/2  
.    1            x^3/2 • x^1/2  
  ——   –   ——————  = 
.   x^1/2              x^1/2 
.    1  –  x^{3/2 + 1/2} 
  ————————  =
.            x^1/2 
.   1  –  x²
  —————  =
.       √x   
.   (1 –  x²) • √x     
  ————————  =  
.       √x • √x 

.  √x  –  x² √x  
  ———————
.         x  
or
.   x^1/2 – x^5/2  
  ———————
.         x 

Answer 2

To get x^2/2, you are adding the exponents in this equation. You cannot add the exponents of the same base unless they are multiplied. For example if the question said:

(x^-1/2)*(x^3/2) = x^{-1/2 + 3/2} = x^2/2 = x

To solve the equation: x^-1/2-x^3/2 you would do this:

x^-1/2-x^3/2 = (1/sqrt(x))-(sqrt(x³))

find a common denominator by multiply (sqrt(x³)) by (sqrt(x))/(sqrt(x)):

(1/sqrt(x))-((sqrt(x³))*(sqrt(x)))/(sqrt(x))

which is equal to:

(1-(sqrt(x³)*sqrt(x)))/(sqrt(x))

assuming x is greater than zero we get:

(1-x²)/(sqrt(x))

and then we can solve for the roots. :)

hope this helps!