Final answer:
To simplify (2x^3y^6)^(3/4), multiply the exponents by 3/4, express in radical form, and extract perfect squares to obtain the final expression 4x^2y^9√y√x in simplified radical form.
Step-by-step explanation:
To express the given quantity, (2x^3y^6)^(3/4), in radical form, we apply the power rule for radicals and simplify using the properties of exponents.
(a) By the power of a power rule, we multiply the exponents inside the parenthesis by ¾: (2^(3×¾))(x^(3×¾))(y^(6×¾)) which simplifies to (2^(9/4))(x^(9/4))(y^(9/2)).
(b) Then, we convert the expression into radical form: √[2^9](x^9)(y^18), where the denominator of the fractional exponent becomes the index of the radical.
(c) Lastly, the expression is simplified by extracting perfect squares where possible: (2^2√[2])(x^2√[x])(y^9), since 2^9 = (2^2)^4×2 and x^9 = (x^2)^4×x.
(d) The final answer in simplified radical form is 4x^2y^9√y√x.