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Operations with Roots
Operations with roots. The test will expect us to do all kind of arithmetic with roots and radical expressions. For example, it might ask us to add or subtract radicals. It might ask us to multiply them or raise them to powers. Each one of these, we will address at some point during this lesson.
I am just listing these as examples to give you an idea. This whole lesson is about doing your arithmetic with radical expressions. So, first of all we will start with addition and subtraction and in fact we will begin with what we cannot do. As we discussed in the previous video, we can not add or subtract directly through the radical signs.
So, for example, if we had to do root 72 minus root 32, it would be incorrect to do subtraction and get square root of 40. It’s an extremely tempting mistake, but it is 100% wrong and mathematically illegal. And it’s very important to appreciate this, because, when your mind is under pressure in the test, your mind is gonna be drawn to making a mistake like this. So it’s especially important to be clear about this, so that you don’t make this mistake when you’re under pressure.
Instead when we have a sum or difference of radicals, we have to simplify each radical separately, by itself. And then we can add or subtract the ones that have the same radical factor. So for example, if we have to subtract root 72 minus root 32, the thing to do is treat each one alone, by itself, and simplify it.
And in fact, this is an other application of simplifying square roots, which we learned in the previous video. Turns out that we use this skill often in problem solving. So, we’ll simplify square root of 72. Course, that’s 2 time 36. Separately we’ll simplify the square root of 32, that is 2 times 16.
And, of course, we can simplify both of those down to 6 root 2 minus 4 root 2. Well, now we have the same radical factor. And, in fact, 6 of anything, minus 4 of that same thing, would have to equal 2 of that same thing. So it has to equal 2 root 2. Notice that we can combine two terms only if they have the same radical factor.
The expression 6 root 2 minus 4 root 3 can not be simplified any further, because those radical expressions are not equal. Now we’ll talk about multiplying dividing radical expressions. Remember first of all that multiplication is commutative and associative. What does this mean? It means when we’re multiplying things we can swap the order around in any way we like.
As long as everything is multiplied together, we can choose which pieces get multiplied together first. We can do them in any order, we have a lot of choice about the ordering and the arrangement of the factors. So suppose we have to multiply two radical expressions.
Now think about this. There’s a lot of multiplication here, 3 times the square root of 5 times 7 times the square root of 2. So really what we have here are four separate factors multiplied together. Well of course these four factors can be grouped in any way. So we’re simply gonna group the whole numbers together, multiply whole numbers by whole numbers.
And group the radicals together, multiply radicals by radicals. So, we’re group the 3 and the 7 together. And separately we’ll group the 5. Root 5 and the root 2 together. Because we multiply right through the radicals. So 3 times 7 is 21.
Root 5 times root 2 is root 10. We get 21, root 10. Sometimes we get a product in the radical that we can simplify. So when we multiple these of course a three times two will give us a six outside the radical. Under the radical we’re going to have five times fifteen, and rather than multiply that I’m going to break the fifteen into factors.
Because when I do that, it becomes very clear I have a 5 times 5. 5 times 5 I can take a square root that. In fact, the square root of that, the square root of 5 times 5 will just be 5. So I can get 6 times 5 times square root of 3, which is 30 root 3. Those were relatively small numbers. Things get trickier here when the numbers are bigger, because the product of radicals can almost always be simplified.
So for example, suppose we had something like, 2 square root of 42 times 4 square root of 63. Of course, the 2 times 4 is 8, that’s easy. Underneath the radical, I’m going to have 42 times 63. Well pause right here. At this point, the absolute worst possible thing you could do would be to find the product of 42 times 63.
So of course if we multiplied 42 times 63, we’d get a number larger than 1,000. Whenever it comes to a point that you have a choice. You can multiply two numbers, and it will wind up with a number larger than 1,000. Rest assured, that performing that multiplication, gratuitously, is the absolute wrong thing to do. It’s much easier to leave things in product form.
And, in fact, it’s gonna be easier to break it down even further to express this in terms of factors. So you don’t wanna get bigger, you wanna break it into small pieces because those are easier to manage. So instead of performing that multiplication, we’re just gonna find the prime factorization of both of those.
Well this makes it very clear that we can pair up some factors. So I’ll just group them like this. The 3 times 3, I can take a square root of that. The 7 times 7, I can take a square root of that. And then of course we have the extra 2 and the extra 3, we can’t simplify those. Those will have to stay under the radical.
But I do get eight times three times seven times root six. Three times seven is 21. We don’t need a calculator for this. Eight times 20 is 160. So 8 times 21 has to be 168. So it’s 168 root six.
Notice, in all of that, we multiplied whole numbers by whole numbers. And separately, we multiplied radicals by radicals, multiplying through the radical signs. One big mistake is to multiply a whole number through a radical sign, to the number on the inside of the radical.
So example, if we add two root six we multiply that to get root twelve. Again, this is the type of mistake its very easy to make when you are under pressure. So it’s very important to pay attention to this. The big idea here is keep separate what’s under the radical, and what’s out in the open as long as you keep those two separate you’ll be fine.
Similarly if we divide we divide whole numbers by whole numbers and we divide radicals by radicals right through the radical signs. So if we have something like this, we separate it out into whole numbers and radicals. As it turns out, 54 is a multiple of 18. 18 times 3 is 54, so that will simplify to 3.
We can multiply, we can divide 35 divided by 5 right through the radical. And we get 3 root 7. Now, as it turns out here, we had an extraordinary coincidence. We have an example in which the radical and the denominator cancelled completely. Now, as it turns out, in general, the radical and the denominator will not cancel.
And we’re going to have to do something with that radical and the denominator. So, that’s a large topic, we’re not going to discuss that in this video. That’s devoted, that is a video all to itself, the video on rationalizing. Which is a couple lessons from that. So the larger topic of how to divide by radicals, we’re just going to put this off and discuss it in great detail in a couple of videos from now.
Finally, we’ll talk about raising radical expressions to powers remember the exponents distribute over multiplication. Thus if we’re squaring a radical expression, we can square the number and the radical separately. So we have 5 root 6 squared. We can just take that exponent of 2 and distribute it to each one of the factors separately.
Well of course we know how to square five and we know how to square root, square the square root of six. The square root of six is the number which by definition if you square it you get six. So, of course, if you square the five, you get 25. If you square the root six, we get six.
Multiply those, we get 150. On harder Quant questions, the test might ask you to raise a radical expression to a higher power. This is less likely, but it could appear in a problem. For example, suppose we’d had 2 root 3 to the power 4. I’m gonna suggest pause the video here, and see if you can figure out what this equals.
Okay. So, one way to approach this is just to apply this exponent of 4 to the two items separately. Applied to the 4, applied to the root 3. Well 2 to the 4th, that’s 16. Root 3 to the 4th, what’s that?
Well, anything to the 4th just means that thing times itself 4 times. So we’ll just write it that way. And of course root 3 times root 3 is 3, so really what I’ll get in that parenthesis is 3 times 3. Which is 9, and then 16 times 9 is 144. So that’s one way to get to the numerical answer.
Another way to attack that same problem is to notice that we could write that 4 as 2 times 2. And the bring only one of the factors of two. Bring that exponent inside the parenthesis. So we’ll have a squared inside and a squared outside.
Well the squared inside, that’s easy. We just talked about how to square things like this. So 2 squared is 4, root 3 squared is 3. That’s 4 times 3 or 12. Now, we can apply the outside squaring, and that gives us 144. That’s another way to approach the same problem.
In fact, that way might be a little more elegant. It’s always good to appreciate cases in which there is more than one way to solve a problem. Keep in mind that any even power of a square root can be written as a power of a whole number. So for example, if we had to deal with square root of 2 to the power of 48.
Well all we have to do is really realize we can write that 48 as 2 times 24 and then bring that 2 inside the parenthesis. So this is root 2 squared, to the power of 24, which means that is 2 to the 24th. We can’t compute the numerical size of that but we still can compare the size of that to something else. For example we can see, that the square root of 2 to the 48th has to be bigger than 2 to the 20th, because that first term equals 2 to the 24th.
In summary, when we add or subtract radical expressions, we simplify each term and combine terms with like expressions. When we multiply or divide radical expressions we can treat the whole numbers and radicals separately. And we can multiply or divide two radicals right through the radical. When we raise the radical expression to how we distribute the exponent to each factor and any even power of a radical is a power of a whole number.
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