Final answer:
To simplify x⁴y⁷/3√(x¹⁰y⁴), we first express the radical as a fractional exponent and then apply the quotient rule for exponents, finally factoring out cubes. The simplest form ends up being y⁵ 3√(x²y²).
Step-by-step explanation:
We are looking to simplify the expression x⁴y⁷/3√(x¹⁰y⁴). To simplify the radical, we note that 3√(x¹⁰y⁴) is the same as (x¹⁰y⁴)^(1/3). Applying the exponent rule (am)n = a(mn), we get x³³³·³³·³³³y⁴·³³·³³³, which simplifies to x³³³⁴y¹³³³´.
Now, dividing the original x⁴y⁷ term by this expression, we apply the quotient rule for exponents: ·x^m/y^n = x^(m-n)/y^(m-n). So, we have x^(4-(10/3))y^(7-(4/3)), which simplifies to x^(2/3)y^(17/3).
Finally, to have positive integers as exponents, we factor out cubes and are left with y⁵ 3√(x²y²), which corresponds to choice D.