Exercise 48 Page 646

To simplify the expression we will start by rationalizing the denominator. We need to make the denominator a perfect square inside the square root. Notice that in our case we already have the perfect square in the denominator.

\frac{9 x ^{3} y}{1 6 x ^{2} y ^{2}}

WritePow

Write as a power

\frac{9 x ^{3} y}{4 ^{2} x ^{2} y ^{2}}

The denominator is already perfect squares only, so we can begin to simplify the quotient. While simplifying, we should consider the index of the radicals to see how we should format our solution.

n a^{n} = {a if n is odd ∣ a ∣ if n is even

Since both radicals are real numbers and the roots are even, the expressions underneath the radicals are positive. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables,

x

and

y .

In the denominator, the index is even and the exponents of $x$ and $y$ are even. Therefore, the expression will be real whether the values of $x$ and $y$ are positive or negative.
In the numerator, the index is even and the exponents of $x$ and $y$ are odd. Therefore, the product of $x^{3}$ and $y$ must be positive in order for this radical expression to result in a real number, so $x$ and $y$ must have the same sign.