Tyler York

June 5, 2023

In this video, Achievable GRE course author Matt Roy explains how to solve a problem that looks quite complicated without even touching the calculator. This video is a great review for multiple radical rules.

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00:00:02 This next problem is a radical problem that looks a bit complicated. But like most problems, if you start off with what you know and try to simplify things, it'll probably look a lot easier. So let's start off with what we do know, which is probably that you know sqrt 64 is 8, so let's fill that in First, we have. 00:00:27 8 -, 2 times the cube root of 8 is all under sqrt. 2 to the 8th power and all of that is inside of a square root. So maybe the next thing we could simplify is the cube root of 8. The cube root is simply asking what number multiplied by itself three times gives you that number. So. 00:00:56 Inside of the house is an 8-2 multiplied by itself three times gives you 8. So on the bottom now we have 8 -, 2 * 2, which is 4, and on the top we have the same as before, so now the bottom simplifies to just four. And let's look at the top for a moment, the square root of anything to a even power. 00:01:27 Is just half of the powers, which is to say that sqrt 2 to the 8th power is just two to the 4th power. Well, there's two reasons for that. You could think to yourself. Well, 2 to the 4th power times 2 to the 4th power is obviously 2 to the 8th power because you add exponents when you. 00:01:50 Multiply 2 numbers with the same base and different exponents. So 2 to the 4th times 2 to the 4th is just the 4 + 4 = 8, so sqrt 2 to the eight must be half of eight, or the exponent must be half of eight. Another way you can think about this though is to simplify the radical into an exponent so 2 to the 8th power is all inside 1/2 power. 00:02:20 Because a square root is, the equivalent is equivalent to 1/2 exponent. And as we already know, exponents outside of exponents mean to multiply, so 8 * 1/2 is just four, so we have two to the 4th power. Well. 00:02:47 Everything is under a radical right now, so we still need to simplify things. 2 to the 4th power we already we. You can guess is 16 because 2 * 2 is 4 and 4 * 4 is sixteen 2 * 2 * 2 * 2 is 16. That's another way of saying that 16 / 4 is definitely 4 sqrt 4 is 2. That's why the final answer, That's why everything written before. 00:03:18 Is actually just equivalent to two.

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