Today, we're going to talk about multiplying radical expressions. Remember, our radical expression is just an expression with a radical sign in it. So we're going to review some multiplication properties, the distributive property, and then how that relates to the FOIL method. And then we'll do some examples, multiplying radical expressions.
So let's start by reviewing our distributive property. Our distributive property tells us that, if I multiply some number, a, by b plus c, that's the same or equivalent to multiplying a and b together, multiplying a and c together, and then adding those two products together. So let's do some numbers to see that that works.
So if I have 3 for my a, 2 for my b, and 4 for my c, I'm going to simplify this by first adding 2 plus 4, which will give me 6. And when I multiply that by the 3, I see that that gives me 18.
So now you should also get 18 on this side. So if I start by multiplying my a and b number together, 3 times 2, and multiplying my a and c number together, 3 times 4, I see that 3 times 2 will give me 6. 3 times 4 will give me 12. And when I add those two products together, I get 18.
So we can see that the distributive property works for these integers or whole numbers. But we also know that the distributive property works for any type of real number, and that includes radicals. So let's do an example with a distributive property involving radicals.
So let's say I have the square root of 6 times 3 plus the square root of 6. Using my distributive property, I'm going to start by multiplying the square root of 6 times 3. And I'm actually going to rewrite this so that the 3 comes in front. And that's generally how you would write a number with a radical that are being multiplied together, just so that you don't get confused about whether this 3 is underneath the radical or not. So I'm going to rewrite this as 3 times the square root of 6.
OK, so I multiplied my first two numbers together. Now I'm going to multiply the square root of 6 times the square root of 6. And I'm going to see where I can simplify.
So I know that, using the multiplication property of radicals, that I can rewrite these two as the square root of 36. Just multiply my two radicands together. And I know that 36 is a perfect square, so that's going to evaluate to be an integer. So the square root of 36 is just 6. And I can't simplify this any more, so my answer is 3 square root of 6 plus 6.
So an important thing to see here is that, when you multiply two square roots together, basically, the two square roots undo each other or cancel each other out. So the square root of 6 times the square root of 6 is just going to give me 6. So if you remember that property or that pattern, then you can skip this step in the middle and just rewrite the product of two square roots with the same radicand as just this single radicand.
So let's do a few more examples multiplying radicals. So let's also review the FOIL method. And the FOIL method is just an extension of the distributive property. We use the FOIL method when we are multiplying two binomials. And it's an acronym for helping us to remember the way that we need to distribute the factors when we're multiplying these binomials.
So FOIL stands for First, Outer, Inner, and Last. And where those words come from are the terms within each of the binomials or sets of parentheses. So here, my first two terms are a and c. My outer terms are a and d. My inner terms are b and c. And my last terms are b and d.
So this is where FOIL comes from. It's just a little easier to remember how you need to multiply or distribute the factors. So let's try it with some numbers.
If I have 8 plus 1 times 4 plus 2, using the FOIL method, I know that I'm going to start by multiplying my first two terms. So 8 times 4 plus my outer two terms, eight times 2, plus my inner two terms, 1 times 4, plus my last terms, 1 times 2. So simplify, I've got 32 plus 16 plus 4 plus 2.
And now, combining these with addition, 32 plus 16 is going to give me 48. 48 plus 4 is going to give me 52. And 52 plus 2 will give me 54.
So if we were to simplify this without using the FOIL method, we can see that we should still get the same answer of 54. So here, in my first parentheses, I would have 8 plus 1, which would give me 9, times 4 plus 2, which would give me 6. And 9 times 6 does give me 54.
So we can see that the FOIL method does work. And we can, again, extend the FOIL method for any type of real number. So let's see how that would work with radicals.
So I've got two binomials multiplied together, so I'm going to use the FOIL method. My first two terms are 3 and 2, so I'll have 3 times 2. My outside two terms are 3 and square root of 4, so plus 3 times the square root of 4. My inside terms are 2 and the square root of 4. And my last two terms are square root of 4 and square root of 4.
Now simplifying this, right away, I see I have two radicands. 3 plus square root of 4 plus 2 square root of 4, it's going to give me 5 square root of 4. Then I see that I can simplify square root of 4 times square root of 4.
I know that the radicals will cancel out, and I'll just be left with 4. And I know that 3 times 2 is going to give me 6. So I can simplify one step farther by adding 6 and 4, so that's going to give me 10 plus 5 times the square root of 4.
For my second example, again, I've got two binomials multiplied, so I'm going to use the FOIL method. My first two terms are 7 and the square root of 3. My outside two terms are square root of 3 and the square root of 5. My inside two terms are 7 and the square root of 5. And my last two terms are square root of 5 and square root of 5.
Now simplifying this, I first see that I've got two radicals that I can simplify using the multiplication property. Square root of 3 times the square root of 5 is going to give me the square root of 15. Bringing down my other terms. Oops. And I also see that square root of 5 times the square root of 5 is just going to give me 5.
So now, I have radicands of 3, 15, and 5. So I know that none of those are like terms, because they're all different radicands. So this is as simplified as my answer can be.
So let's go over our key points from today. As usual, make sure you have them in your notes so you can refer to them later. We saw that the distributive property in the FOIL method can be used with radical expressions. And a helpful hint is to remember that multiplying 2 square root terms together will cancel out the square root operation. So for example, the square root of 8 times the square root of 8 is just 8.
So I hope that these key points and examples helped you understand a little bit more about multiplying radical expressions. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks, for watching.
Key Formulas
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.