assume that all variable are positive real numbers

Question

3^-1/3 x^2/5 y^-5/8

______________

3^-10/3 x^-5/2 y^3/8 

 

This is a fraction

Answer 1

When dealing with exponents in equations with fractions the simplest way to think of it is

that if we move any unit from the numerator to the denominator (top to bottom) or from the

denominator to the numerator (bottom to top) we simply change the sign of the exponent. 

For example x² = 1/x^-2 ..,  a^-3/b^-4 = b⁴/a³, etc..
 
You can simply move all the denominator elements up top, changing their exponent signs and
then multiply across to see what you get.

(3^-1/3x^2/5y^-5/8)/(3^-10/3x^-5/2y^3/8) = (3^-1/3)(3^10/3)(x^2/5)(x^5/2)(y^-5/8)(y^-3/8)

 

= (3^-1/3+10/3)(x^2/5+5/2)(y^-5/8-3/8) = (3^9/3 )(x^4/10+25/10)(y^-8/8)

 

= 3³x^29/10y^-1 = 27x^29/10y^-1  

 

If you wish all ending exponents to be positive then move y^-1 down creating a new fraction:

 

(27x^29/10)/y

Answer 2

To divide like bases with exponents, subtract the exponent of the denominator from the exponent of the numerator.

 

3^-1/3-(-10/3) x^2/5-(-5/2) y^-5/8-(3/8)

 

3^-1/3+10/3 x^4/10+25/10 y^-5/8-3/8

 

3^9/3 x^29/10 y^-8/8

 

3³ x^29/10 y^-1