Which of the following gives `((x^(2)y)^(3))/(x^(7)y^(2))timessqrt(sy)` written in simplest form?

Options

A. `(ysqrt(y))/(sqrt(x))`
B. `(3sqrt(y))/(sqrt(x))`
C. `sqrt((3y)/(y))`
D. `(y)/(sqrt(xy))`

Answer

A

Explanation

Difficult: Hard
Category: Passport to Advanced Math/ Exponents
Strategic Advice: Before you can combine radical expressions with non-radical expressions, you must write them in the same form. Here, you can rewrite the square root of xy as `(xy)^((1)/(2))`
Getting to the Answer: First, simplify as much as possible using the rules of exponents. Then rewrite the radical as a power of xy and simplify further, using the rules of exponents to combine terms that have the same base:
`((x^(2)y)^(3))/(x^(7)y^(2))sqrt(xy)=(x^(6)y^(3))/(x^(7)y^(2))sqrt(xy)`
`=(y^(3-2))/(x^(7-6))sqrt(xy)`
`=(y)/(x)sqrt(xy)`
`=x^(-1)y^(1)sqrt(xy)`
`=x^(-1)x^((1)/(2))y^(1)y^((1)/(2))`
`=x^(-1+(1)/(2))y^(1+(1)/(2))`
`=x^((1)/(2)y^((3)/(2))`
The answer choices are given as radicals, so change the powers back to roots. The negative power on x moves it to the denominator, and for the y term, use the saying "power over root" to help you write the correct radical form:
`x^((1)/(2))y^((3)/(2))=(y^((3)/(2)))/(x^((1)/(2)))=(sqrt(2)(y^(3)))/(sqrt(x))=(ysqrt(y))/(sqrt(x))`
The answer is (A).