Working with Square Roots
Episode #5 of the course Foundations of mathematics by John Robin
We met square roots in Lesson 2, and today, we’re going to spend more time here because there’s a lot we can do with them.
Multiplying Square Roots
What is √2 x √8? It turns out the rule is very simple and exactly what your intuition might tell you: Simply multiply the numbers together and group them together under one square root.
So, here: √2 x √8 = √(2×8) = √16.
You might recognize 16 as a perfect square—that is, 16 is also 4×4. Perfect squares are numbers that are formed by multiplying together an equal pair of natural numbers. For example, 2×2 makes 4, 3×3 makes 9, 4×4 makes 16, 5×5 makes 25, and so on. This is handy for working with square roots because if you can spot a perfect square, you can calculate the square root without needing a calculator.
Dividing Square Roots
What is √2 / √8? Fortunately, the rule is simple here, too, and exactly what you’d expect: You can just divide 2 by 8 and put them together under one square root:
√2 / √8 = √(2/8) = √(1/4)
And just as we can combine our two square roots into one by dividing the top number by the bottom inside, we can also separate a fraction inside a square root into the square root of the top number divided by the square root of the bottom:
√(1/4)= √1 / √4
We can evaluate each of these separately now: √1 = 1, since 1 = 1×1. And you may recall from above that 4 is a perfect square: 2×2. So, the fraction now simplifies to: 1/2.
There we go. Using the rules for division, we can evaluate a fraction made of two square roots, √2 / √8, and we find it is 1/2.
Adding Square Roots
You can also break the square root of a product of numbers (remember that means numbers multiplied together) into a product of the square root of those numbers. You’ll see this in action as we move to adding square roots.
What is √2+ √8? Observe the following:
√2 + √8 = √2 + √(2×4) = √2 + √2 x √4 = √2 + 2x√2
Above, we had √8, and it was useful for us to break that down. Following our factoring routine, we divide by the smallest prime, 2. This turns into 2×4. We could have gone further, 2x2x2 (or 23 as we learned in Lesson 3), but when dealing with square roots, your goal is to factor so you uncover perfect squares. This allows us to separate the product into two square roots, and one of them will become a whole number.
We are almost done. Now, notice that √2 and 2x√2 have one thing in common: √2. There is 1 √2 on the left and 2 √2s on the right. This means there are a total of 3 √2s altogether: √2 + 2x√2= (1+2)x√2 = 3x√2.
And that’s how we add together square roots.
Subtracting Square Roots
The rule is identical to adding square roots. Let’s try this on another example:
√2 – √18
First of all, we can factor 18 into 2(9). Notice that when I did this, I was hunting for a perfect square, in this case, 9, which is 3×3.
√2 – √(2×9) = √2 – √2x√9 = √2 – 3x√2
Now we look at what these two have in common: √2. The left root has 1 √2 and the right has 3 √2s. We are subtracting, so we have 1 – 3, which is -2 √2s:
√2 – 3x√2 = (1-3)x√2 = -2x√2
And that’s it for our tour of square roots. Get ready for the next step tomorrow, where we’ll combine fractions and square roots together.
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