Simplify the expression x^1/4 / x^−4/5x Assume x>0

Question

Simplify the expression x^1/4 / x^−4/5x Assume x>0

 The answer can be written in the form 1x^b where 

the exponent b is: 

Answer 1

(I will start by saying that I am making an assumption that the x after the -4/5 is a typo. If it is not, we need to see if we can clarify where that third x is in the problem; it's difficult to tell where it would be in this linear format.)

To simplify this, remember that when we have an expression x^a/x^b, we can simplify it, and keep the variable in the numerator, by writing it as x^a-b. In the case of this problem, then:

x^(1/4) / x^(-4/5) = x^{(1/4 - (-4/5))}

Simplifying the exponent requires finding a common denominator for the fractions (20) and rewriting each fraction as a fraction with this denominator.

1/4 = 5/20

4/5 = 16/20

5/20 - (-16/20) = 5/20 + 16/20 = 21/20

Therefore:

x^(1/4) / x^(-4/5) = x^(21/20)

The exponent, b, then, is 21/20.