^4sqrt symbol(x^2 y^3)^3sqrt symbol(xy)
To simplify the expression ^4√(x^2y^3)^3√(xy), we can follow these steps:
Step 1: Simplify each radical separately.
The ^4√(x^2y^3) can be simplified by raising each factor inside the radical to the power of 1/4:
^4√(x^2y^3) = (x^2)^(1/4) * (y^3)^(1/4)
= x^(2/4) * y^(3/4)
= x^(1/2) * y^(3/4)
Similarly, the ^3√(xy) can be simplified by raising each factor inside the radical to the power of 1/3:
^3√(xy) = (x)^(1/3) * (y)^(1/3)
Step 2: Combine the simplified radicals.
Now, we can combine the two simplified radicals by multiplying their respective terms:
(x^(1/2) * y^(3/4)) * (x^(1/3) * y^(1/3))
Step 3: Apply the laws of exponents.
To simplify the expression further, we can use the laws of exponents to add the exponents of like variables:
x^(1/2) * x^(1/3) = x^((2/6) + (1/3)) = x^(5/6)
y^(3/4) * y^(1/3) = y^((3/4) + (1/3)) = y^(13/12)
So the simplified expression is:
x^(5/6) * y^(13/12)