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Lesson 122: More About Square Roots (Radicals) (Part 1 of 2)
We've done some work with square roots in earlier lessons. In this lesson we'll learn some special things we can do with them.
First, note that the square root sign is often referred to as a radical. The number under the sign is called a radicand. Be sure to remember these terms, as they come up very frequently. We are frequently asked to simply radicals. Here is what we do. Look at √48. To simplify it, we have to find factors of 48 which are perfect squares. Review the lessons on factors and perfect squares if you're not familiar with these concepts. It's important to memorize as many perfect squares as you can, such as 4, 9, 16, 25, 36, 49, etc. Memorizing these will help you when you work with radicals.
What we do is start listing perfect squares, and then see which ones divide into 48. We want the largest one. We'll start with 4. That divides into 48, but let's see if there is a larger one. 9 does not divide. 16 divides evenly into 48, and we'll use that one, since it's the largest. What we do is split 48 into two of its factors: 16 and 3. We'll write √(16·3).
Now we have to introduce a rule. When we have the square
root of a product, we can rewrite it as the product of the square
roots of each factor. In symbols, we can say that √(a·b) = √a · √b. In words, we can say that
the square root of a product is the product of the square roots. Getting back to our example
above, we can write √16 · √3. Now we can simplify √16 by
writing 4, giving us a final simplified answer of 4√3.
We can also take the square root of a fraction (a division
problem). We have a
rule for that: √(a/b) = √a / √b. In words, we can say that the
square root of a quotient is the quotient of the square roots.
Let's try simplifying √(25/49). We can write √25 / √49, which
simplifies to 5/7.
It's very important to understand that each square root has both a positive and a negative solution. Let's look at √64. This is asking us to find a number, that when multiplied by itself, will equal 64. One answer is 8, but there is another answer. -8 times -8 also equals positive 64. That means that 8 and -8 are both answers. Usually you would just give the positive answer (the principal root), unless you were told otherwise. However, the "official" answer can be written as ±8, read as "plus or minus 8."
Note that if the problem specifically wants you to provide the negative square root, it will be written as -√a. That means to give the negative answer.
You'll have much more practice with radicals in later lessons, and in the next lesson you'll learn about how to do basic operations with radicals.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.