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Simplifying Roots

Simplifying roots. This is a very important topic on the test. So, often when we have to take the square root of a problem, square root in a problem, the problem itself will result in the square root of a larger number, say the square root of 75 or the square root of 80, something like this. But the square root in that form will not appear among the answer choices.

In other words, we’ll do all our math correctly, and yes, square root of 80 should be the correct answer, but we will not see square root of 80 listed as an answer choice in the multiple choice answers. What’s going on here? The answer choice will be in simplified form, and we have to recognize how to simplify the radical.

So that’s why this is a very important skill. As we learned in the previous lessons, roots distribute over multiplication, so we can separate the root of the, of a product into the product of the roots. That’s a really big idea, we’ll return to that in a minute. First of all, remember that of course, it’s easy to find the square root of perfect squares.

As a reminder here, the square roots of the first 15 perfect squares, these are good numbers to know. And of course, if you know the first 15 perfect squares, it is very easy to find their square roots. Now as we learned in the previous lesson, roots distribute over multiplication. So we can separate the root of the product into the product of the roots, in other words, the root of P times Q equals square root of P times square root of Q.

We can separate it by a multiplication. If either P or Q is a perfect square, then that square root would be very easy to simplify, and this would make the whole thing simpler. For example, suppose at some point in a problem, we needed the square root of 75, this was the hypothetical problem I, I expressed earlier. We do all our correct math, we find square root of 75 is the answer.

It’s not listed among the answer choices. What’s going on? Well, clearly, 75 is a multiple of 25 and 25 is a perfect square and we can use this to our advantage. What I’m gonna do is I’m gonna start with square root of 75 and I’m gonna write that 75 as 25 times 3 because of course, 75 is 25 times 3.

I’m gonna separate that out into two separate roots and of course the square root of 25, that’s something I can simplify. That’s a perfect square, so square root of 25 is just 5 and I can just write this as 5 times root 3, that is simplified. That last form is the simplified form of the square root of 75. The form radical 75 would never appear as a multiple choice answer choice.

It would always appear in the simplified form, 5 root 3. So this is one of the reasons it’s very important to know how to simplify. I’ll explain this again in more algebraic terms now. Suppose N is the number under the radical. If N has as one of its factors, a perfect square P, then we can replace N with P times Q for some Q.

Take the square root of the the product separate and of course, the square root of the perfect square P will come out evenly and this will simplify the entire root. For example, 12 is divisible by a perfect square. 12 is divisible by 4, so we write this as 4 times 3, we take the square root of 4 and of course, that’s 2. So we get 2 root 3.

63 is divisible by 9. So we can write this as square root of 9 times 7. The first square root, square root of 9 becomes 3. So I get 3 root 7. And the square root of 80, we mentioned this earlier. 80 is actually divisible by 16.

So it’s 16 times 5, take the square root of 16 and we get 4 and this is 4 root 5. So these are examples of simplified square roots. So first of all, it’s important to be able to do this so that you can recognize the right answer in the form in which it appears in the answer choice. Also, as we’ll see, it’s important to do this even as you’re working through the problem because obviously, when you simplify things, you’re working with smaller numbers and it’s always easier to work with smaller numbers.

Suppose the number beneath the radical is particularly large, suppose we had to find the square root of 2,800. It certainly will help to fact out, factor out large squares such as 100. It may be necessary to find the full prime factorization of the number. I’ll talk about square roots and prime factorizations in a minute. Right now we can factor out 100, so why don’t we just do that?

We’ll express that 2800 as 28 times 100. The square root of 100 is very easy, that just becomes 10. So we have 10 times the square root of 28. This is not fully simplified because 28 itself is divisible by 4. So we’ll write that as 4 times 7. Square root of the 4 is 2.

So that part under the radical, that becomes 2 root 7. Multiply by 10 and I get 20 root 7. That is the simplified form of radical 2800. In the rare instance that we were given the prime factorization of a particular, a particular large number, it would be very easy to find the simplified square root.

So they’re really handing you a gift on a silver platter if they give you the prime factorization of a number, and they say find the square root of this number. So, that’s something very important to appreciate here. Every pair of prime factors and in fact, every even power of a prime factor is a perfect square and can be simplified. So, here is a practice problem, pause the video, and then we will talk about this.

Okay, so we have a bunch of different powers. I’m gonna take a square root and apply the square root to each individual set, each individual power. Now let’s see, that 2 to the 6, that’s an even root. I could write that as 2 to the 3rd times 2 to the 3rd. That’s what the laws of exponents tell us, 2 to the 3rd times 2 to the 3rd is 2 to the 6th.

So in other words, 2 to the 3rd squared is 2 the 6th. So I can write that as 2 to 3rd times 2 to the 3rd. 3 to the 5th is not even, so I’m gonna write that as 3 squared times 3 squared times 3 to the first. So at least, there’s a square in there. The 5 squared, that’s already a square, square root of 7, we can’t do anything with that.

So the first term is gonna simplify the square root of 2 to the 3rd times 2 to the 3rd, that’s gonna simplify as just 2, 2 to the 3rd. The next one is gonna simplify to 3 squared times the square root of 3, the square root of 5 squared is 5, and then radical 7, we’re just stuck with that. We have to keep that as is.

Well now we can multiply a little bit. First of all, we know of course that 2 cubed is 8, 3 squared is 9. I’m gonna multiply that 8 times 5 to get 40, and then 40 times 9 is 360 and that 3 times 7 under the radical is 21, so we get 360 radical 21 and that is the simplified form of the square root of n. In summary, we simplify square roots by factoring up the largest perfect square factor.

If we find the prime factorization or are given it, then we can use that. Any pairs of prime factors and any even powers of prime factors are perfect squares, and we can simplify those square roots.

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