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

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1 Answer

(x2z)1/4/x-1/2z1/2
 
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
 
x2*1/4z1/4/x-1/2z1/2
 
x1/2z1/4/x-1/2z1/2
 
To further simplify from this point, we make use of the fact that xA/xB=xA-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.
 
x1/2-(-1/2)z1/4-1/2
 
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/z1/4