Simplify the following. ( 2 5 ) ( 4 7 ) (2\sqrt5)(4\sqrt7)

Welcome back, everyone. So let's take a look at a couple more practice problems in our practice set. We've already seen a couple of different situations of multiplication of radical expressions. Let's do a few more. The idea here is you should try these on your own. And if you get the wrong answer, that's okay. I'm going to walk through this step by step and maybe you'll understand where you went wrong a little bit. Alright? Let's Let's get started. We've got two radical five times four radical seven. We're supposed to multiply and simplify the following. So here's what happens. We've got these two expressions that are multiplied, but I don't see a plus or minus sign anywhere, which means that I don't have to do any kind of distribution or the distributive property. But all I can do here is I can just multiply these two radical expressions. And whenever that happens and you don't have pluses or minus signs, you can kind of just pretend that the parentheses will just go away and you can start to multiply the non radicals like two and four and then the radicals like radical five and radical seven. Because multiplication in the order doesn't matter, then you can kind of just rearrange these things and just multiply the non radicals with the non radicals and then vice versa. So here's what happens. If I multiply the two non radicals, this is two times four. And if I multiply the two radicals, we have radical five times radical seven. We know that we can use the product rule of radicals to combine these under one larger radical. So So what happens is the two times four just becomes eight, and then the radical five times radical seven will just combine to radical 35. All right. Now, whenever you get an answer like this, where you have eight radical five, you have a mixture of non radicals and radicals, You should always look at the number that's inside the radical and figure out if there's any further way that you can simplify this. Usually with a larger number like this, you're going to try to think of two numbers that multiply to this number, in this case, 35, in which one of them is a perfect square because then you can pull it out and then you can further simplify and reduce this number. So let's just think about this. Are there any factors or there any two numbers in which one of them is a perfect square? Well, the only factors of 35 are one times 35. That doesn't work because it doesn't simplify a radical. Two doesn't work. Three doesn't work. Four doesn't go into 35. The only other two numbers that work here are five times seven, but that also doesn't really work here either because neither five nor seven are perfect squares. So What actually happens here in this problem is that there's actually no way that you can reduce this eight radical 35 into anything simpler. So this actually is your final answer. Eight thirty five is the simplest way that you can write this expression. Alright, folks, that's all there is to it. Let me know if you have any questions and I'll see you in the next

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44m

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Multiplying, Dividing, and Rationalizing Radicals

Intermediate Algebra

9. Roots, Radicals, and Complex Numbers

Multiplying, Dividing, and Rationalizing Radicals

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Multiple Choice

Simplify the following.
$(2 \sqrt{5}) (4 \sqrt{7})$

$8 the square root of 2$

$8 the square root of 13$

$6 the square root of 35$

$8$\sqrt{35}$$

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Verified step by step guidance

Start with the expression to simplify: $ (2\sqrt{5})(4\sqrt{7}) $.

Use the property of multiplication for radicals: multiply the coefficients (numbers outside the square roots) together and multiply the numbers inside the square roots together separately. This gives: $ (2 \times 4)(\sqrt{5} \times \sqrt{7}) $.

Multiply the coefficients: $ 2 \times 4 = 8 $, and multiply the radicands inside the square roots: $ \sqrt{5} \times \sqrt{7} = \sqrt{5 \times 7} = \sqrt{35} $.

Combine the results to get: $ 8\sqrt{35} $.

Since $ \sqrt{35} $ cannot be simplified further (35 has no perfect square factors other than 1), the expression is fully simplified as $ 8\sqrt{35} $.

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