Simplify the radical expressions in Exercises 58 - 62. 4∛16 + 5∛2

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Accepted Answer

Hello today we're gonna go ahead and simplify a radical expression. Now, before we start simplifying things, let's go ahead and break down each of the radical expressions into a product of prime numbers. So starting with 405, We can go ahead and break this down to five times 81. Five is a prime number. So I'm gonna go ahead and circle it. But 81 could be broken down to nine times nine, which can both be broken down to three times three. Now, that means that we have four instances of three and one instance of five plugging this back into the 1st. 4th root is going to give us three times the fourth root of 3 to the Power four times 5. Now we're going to repeat this process for the next 2/4 roots. So moving on to the next 1 80 could be broken down to 10 times 8. 10 could be broken down to five times two, which are both prime numbers and eight could be broken down to four times two and four could be broken down to two times two. So we have more instances of two here. So in total we have four instances of two and one instance of five plugging this back into the fourth root is going to give us two times the fourth root of two times itself, four times or two to the fourth. Power Times five. And finally we're gonna go ahead and break down 1280. Doing so we can go ahead and break it down to two times 640 two is a prime number 640 could be broken down to eight times 80 80 could be broken down to 10 times eight. 10 will be broken down to five times two and eight will be broken down to four times two. Go ahead and circle the prime numbers we have so far, and four will be broken down to just two times two. Mhm No to continue this eight again will be broken down to four times two and four will be broken down to two times two. So in total, we have 12345678 instances of two and one instance of five. Putting this back into the fourth root is going to give us the fourth root of 2 to the power of eight times 5. Now let's go ahead and put everything back together. Doing so is going to give me three times the 4th root of three to the power of four times five plus two times the fourth root of two to the fourth, times five minus the fourth root Of 2 to the 8th. Power Times five. Now let's go ahead and simplify these a little bit three to the fourth. Power inside the fourth root is just going to give me three and we can't simplify the fourth root of five. So we're going to leave that on its own. So now we're gonna have three times three times the fourth root of five. Next we can simplify two to the fourth power inside the fourth root because that's just going to leave us with too. And just like before we can't simplify five in the fourth root. So we're gonna leave that alone. So what will be left with is two times two times the fourth root of five. Next we're gonna go ahead and simplify the last 12 to the eighth inside the fourth root will be reduced to two squared and just like before we can't reduce the fourth root of five, so we're going to leave it as is so this will be multiplied by the fourth root of five. Now let's go ahead and multiply all of our constants together. So three times three is going to be nine times the 4325, Two times two is going to be four times to 4, 3-5 and two squared is going to give us four times the fourth root of five. Now we can go ahead and combine these three together because all of them have a common term of the fourth root of five. But if you notice We have a positive four through to five and a negative 4, 3-5 and both of these are going to go ahead and cancel each other out. That means that the only value will be left with is nine times the fourth root of five and that's going to be our solution That also tells us that the solution to this problem here is going to be D. So I hope this video helps you simplify radical expressions, and I'll go ahead and see you all in the next video.

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