Simplify the expression. Write the answer with positive exponents. Assume all variables represent positive numbers

Question

Simplify the expression. Write the answer with positive exponents. Assume all variables represent positive numbers. 

(x²z)^1/4/x^-1/2z^1/2

Answer 1

(x²z)^1/4/x^-1/2z^1/2

 

First we want to simplify the expression in the numerator. Since the term (x²z) is being raised to the 1/4 power, we multiply the exponents of each variable to give us a simplified power. The denominator is already simplified. This gives us

 

x^2*1/4z^1/4/x^-1/2z^1/2

 

x^1/2z^1/4/x^-1/2z^1/2

 

To further simplify from this point, we make use of the fact that x^A/x^B=x^A-B. That is, the power of exponential terms divided by terms with the same base is equal to the difference in the exponents. We do this for both x and z.

 

x^1/2-(-1/2)z^1/4-1/2

 

We recognize that the subtraction of a negative term in a power of x leads us to add the two terms, and we find a common denominator, 4, to subtract the powers of z. We find

 

x¹z^-1/4

 

The power of 1 in the x term gives us our base, x, and the power of -1/4 in the z term can be rewritten as a positive exponent in the denominator of our simplified expression. This gives the answer

 

x/z^1/4