Use the product rule to rewrite the term inside the radical as a product, then simplify. 180 \sqrt{180}

Here we're asked to use the product rule to rewrite the term inside of the radical as a product, and then simplify. We're looking here at the square root of 180. Now remember back in our lesson, we rewrote 50 under our radical as the product of a perfect square and another value in order to use radical as the product of a perfect square and another value in order to use the product rule to simplify. That's the same thing that we want to do here. And the way that we do that is we write out all of the factors of our numbers since this is a much larger value. And what we're looking for is the largest factor that is a perfect square. So thinking about the factors of 180, I have one times 180, two times 93 times 64 times 45, five times 36, six times 30. Seven is not a factor. Eight is not a factor either. I have nine times 20 and then 10 times 18. So I'm looking at all of these factors. I wanna look for the largest one that is a perfect square. So in this four times 45, I know that four is a perfect square, but in my very next line, I have five times thirty six and thirty six is a larger perfect square. Then later on, I have nine times 20, so that perfect square of nine, but that is not a larger value than 36. So this is the factor pair that I want to use here in order to simplify this radical. So the square root of 180, I'm going to rewrite this as the square root of five times 36. Now I can use the product rule to split this into two radicals. So that becomes a square root of five times the square root of 36. Now the square root of 36 is six. It's a perfect square. So that makes this the square root of five times six. And when we write constants with radicals, we want to write that constant first. So rewriting this as six root five is gonna give me my solution here. Now as we continue on, I'm gonna give you even more guidance on how exactly we wanna think about rewriting terms under a radical as a product in order to simplify later on. We're gonna see this come up a lot in future lessons. For now, we at least know how to do it for these basic radicals, and we can continue getting practice. I'll see you in the next one.

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Use the product rule to rewrite the term inside the radical as a product, then simplify.180\(\sqrt{180}\)

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Use the product rule to rewrite the term inside the radical as a product, then simplify.
$the square root of 180$