9.5 Divide square roots  (Page 2/4)

If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.

Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator , we write an equivalent fraction with a rational number in the denominator.

Let’s look at a numerical example.

$\begin{array}{cccc}\text{Suppose we need an approximate value for the fraction.}\hfill & & & \frac{1}{\sqrt{2}}\hfill \\ \text{A five decimal place approximation to}\sqrt{2}\text{is}1.41421.\hfill & & & \frac{1}{1.41421}\hfill \\ \text{Without a calculator, would you want to do this division?}\hfill & & & 1.41421\overline{)1.0}\hfill \end{array}$

But we can find a fraction equivalent to $\frac{1}{\sqrt{2}}$ by multiplying the numerator and denominator by $\sqrt{2}$ .

Now if we need an approximate value, we divide $2\overline{)1.41421}$ . This is much easier.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.

Similarly, a square root is not considered simplified if the radicand contains a fraction.

To rationalize a denominator, we use the property that ${\left(\sqrt{a}\right)}^2=a$ . If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

Simplify: $-\frac{8}{3\sqrt{6}}$ .

Solution

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by $\sqrt{6}$ .

Multiply both the numerator and the denominator by $\sqrt{6}$ .
Simplify.
Remove common factors.
Simplify.

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Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.

Simplify: $\sqrt{\frac{5}{12}}$ .

Solution

The fraction is not a perfect square, so rewrite using the Quotient Property.
Simplify the denominator
Rationalize the denominator.
Simplify.
Simplify.

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Simplify: $\sqrt{\frac{11}{28}}$ .

Solution

Rewrite using the Quotient Property.
Simplify the denominator.
Rationalize the denominator.
Simplify.
Simplify.

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