Exercise 44 Page 371

To rationalize the denominator, we will multiply the numerator and denominator by a factor that will make the denominator a perfect square inside the square root. We will do this using the fact that we can multiply the radicands of radicals if they have the same index.

If n a and n b are real numbers, then n a \cdot n b = n a b .

Let's start by finding the exponents necessary to create perfect squares in the denominator. Our goal is to have two of each factor.

\frac{5}{8 x}

WritePow

Write as a power

\frac{5}{8 ^{1} x ^{1}}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 8 ^{1} x ^{1}}{b \cdot 8 ^{1} x ^{1}}$

\frac{5 \cdot 8 ^{1} x ^{1}}{8 ^{1} x ^{1} \cdot 8 ^{1} x ^{1}}

ProdSqrt

$a \cdot b = a \cdot b$

\frac{5 \cdot 8 x}{8 ^{1} x ^{1} \cdot 8 ^{1} x ^{1}}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{5 \cdot 8 x}{8 ^{2} x ^{2}}

Multiply

\frac{4 0 x}{8 ^{2} x ^{2}}

Now that we've found the factors that will make the radicand of the denominator perfect squares only, 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 variable

x .

In the numerator, the index is even and the exponent of $x$ is odd. Therefore, in order for this radical expression to result in a real number, $x$ must be positive.

This means that if we remove