Final answer:
The expression sqrt(dfraczz^scriptsizedfrac13) simplifies to z^(1/6), which can be evaluated to an integer, fraction, or an exact decimal if the value of z is known.
Step-by-step explanation:
The expression provided is sqrt(dfraczz^scriptsizedfrac13), which can be simplified using the properties of exponents. To rewrite this expression as an integer, fraction, or an exact decimal, we will first simplify the expression within the square root.
Let's recognize that z to any power can be written with a fractional exponent. For example, z^n can become z^(1/n) if we are taking the nth root of z. Here, we are taking the square root, which can be represented by the exponent 1/2. So, sqrt(z) equals z^(1/2).
Next, we simplify the denominator: z^scriptsizedfrac13 can be written as z^(1/3). Now let's write the complete expression with these exponents:
sqrt(z)/z^(1/3) becomes z^(1/2) / z^(1/3).
Applying the laws of exponents, we subtract the exponents when dividing:
z^(1/2 - 1/3) = z^(3/6 - 2/6) = z^(1/6).
Therefore, the answer is z^(1/6). This represents the expression in simplified form, which may be properly evaluated to an integer, fraction, or an exact decimal, given a specific value for z.