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Lesson 123: More About Square Roots (Radicals) (Part 2 of 2)
We've done some work with square roots in earlier lessons, and in the previous lesson. In this lesson we'll learn some more special things we can do with them.
First, review the previous lesson and make sure that you fully understand it. In this lesson we'll learn how to do basic operations with radicals. The first thing we need to learn is how to add and subtract radicals. Let's start with a simple example:
7√5 + 6√5. Since the radicals are the same, all we need to do is add the coefficients (the numbers before the radicals). We get 13√5. Think of it as adding 7 apples and 6 apples. Since we're adding the same things, we can just add in the usual way. If the radicals are different, then we cannot add them. It would be like adding apples and oranges. If you are able to simplify one of the radicals like we learned in the previous lesson, then it's possible that you'll end up with radicals that are the same, and then you could add.
Subtraction is done the same way. We can do 9√17 - 2√17 and get 7√17, since the radicals are the same.
Multiplication is very easy, and we even learned the rule for it in the previous lesson. Recall that we could take something like √45, and break it up into √(9 · 5), which in turn could be split up as √9 · √5.
What that means it that if we want to multiply two radicals which have different radicands, we can just multiply the radicands, and put the product under a radical sign. For example, to multiply √8 times √3. We just multiply 8 times 3 to get 24, and we get √24. If we wanted to, we could then simplify our answer like we learned in the previous lesson.
If there are coefficients in front of each radical, just multiply the coefficients separately from the radicals. For example, let's do 4√3 · 7√5. We'll multiply the coefficients to get 28, and the radicands to get 15. Our answer is then 28√15.
Division is similar, and again, we learned the rule in the previous lesson. To divide radicals, the radicands do not need to be the same. We just do the division, and put the quotient under a radical sign. For example, to do √60 ÷ √4, we just do the division of the radicands to get 15, and we put it under a radical sign as √15.
Just like with multiplication, if the are coefficients in front of each radical, just divide them separately. For example, 27√60 ÷ 3√4 = 9√15.
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