Radical Expressions, Combining Fractions and Square Roots

Episode #6 of the course Foundations of mathematics by John Robin

 

We’ve worked now with square roots and know how to put two of them together. But what happens when we mix square roots and fractions? I’m talking about something that looks like this: √5 + (7 / √5). (Run and hide is not an option.) Fortunately, we’ve learned enough in the first half of our course to be able to handle this, and today, I’m going to equip you to handle any kind of expression that looks like this.

These are called radical expressions. Any expression that has both fractions and square roots in it is considered radical. Our goal when we see a radical expression is to combine our terms into one fraction that does not have a square root in the denominator.

Let’s learn through doing, using our above example, √5 + (7 / √5).

1. Get rid of all square roots in any denominator by multiplying a given fraction with √(denominator) in its denominator by √(denominator) / √(denominator):

In this expression, there is only one square root in the denominator. It’s in the right fraction:

7 / √5

The √(denominator), then, in this case is √5. So, we multiply that fraction 7 / √5 by √5 / √5:

(7 / √5) x (√5 / √5)

= (7 x √5) / (√5 x √5)

This has the effect of cancelling the √5 in that fraction’s denominator, since √5 x √5 = 5. So, our expression now becomes:

√5 + 7/5 x √5

2. Once all fractions are free of square roots in the denominator, find a common denominator:

Recall from Lesson 2 that √5 is a fraction, namely the fraction √5 / 1. This is because every number written with no denominator can be assumed to have the denominator 1.

This means we now have two fractions, the left with denominator 1 and the right with denominator 5. We dealt with how to find a common denominator in Lesson 4:

[Numerator left]/[Denominator left] x [Denominator right]/[Denominator right]

+

[Numerator right]/[Denominator right] x [Denominator left]/[Denominator left]

In this case, it becomes:

√5 / 1 x 5/5 + (7x√5) / 5 x 1/1

= (5x√5) / (1×5) + (7x√5) / 5

= (5x√5) / 5  + (7x√5) / 5

You might have noticed that on the right, the fraction 1/1 disappeared. I included this only to show you how we apply this rule generally for any fraction, but of course, when you practice, you’ll learn to skip this step, since multiplying by 1 has no effect.

We’re almost done. Now we can apply what we learned in Lesson 4:

3. Follow the rules of fraction addition:

a. Make sure the fractions have the same denominator (which is why we did the above steps).

b. Add the two numerators, and combine into one fraction with the common denominator.

So, (5x√5) / 5 + (7x√5) / 5 becomes:

(5x√5 + 7x√5) / 5

One last step …

4. Follow the rules of square root addition/subtraction if there are any square roots in the numerator:

Yesterday, we learned how to handle something like 5x√5 + 7x√5. We found what they had in common (in this case, √5), then added together (or subtracted) the numbers in front. Here, we have 5 √5s on the left and 7 √5s on the right, so we have 5+7 = 12 √5s total.

Our expression now becomes (12x√5) / 5.

The last thing we might need to do is simplify. In this case, 5 is a prime number and 12 = 2²(3), so nothing will divide out. We are done.

You can apply this procedure with any radical expression to convert it into one fraction with no square root in the denominator.

Tomorrow, we’ll move to our next stop: exponents.

 

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