Simplifying radicals (higher-index roots)
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Multiply and simplify 5 times the cube root of 2x squared times 3 times the cube root of 4x to the fourth. So the two things that pop out of my brain right here is that we can change the order a little bit because multiplication is both commutative-- well, the commutative property allows us to switch the order for multiplication. And so we can get the constant terms. We can multiply the 5 times the 3. And then the other two things that we're multiplying-- they're both the cube root, which is the same thing as taking something to the 1/3 power. So the cube root of x-- this is exactly the same thing as raising x to the 1/3. So let's do that. Let's switch the order and let's rewrite these cube roots as raising it to the 1/3 power. So I have the 5 and the 3. So that's going to be 5 times 3. And then we have the cube root of-- do that in a new color. Then we have the cube root of 2x squared. So this I can rewrite as 2x squared to the 1/3 power. And then I have the cube root of 4x to the fourth. So that's the same thing as 4x to the fourth to the 1/3 power. And now we know from our exponent properties, if we have two things that are both raised to the same power and then we take their product, we could just take their product first and then raise it to the power. So if I have a to the x power times b to the x power, this is the same thing as a times b to the x power. So we can simplify this part of the expression right over here as 2x squared times 4x to the fourth to the 1/3 power. And of course, 5 times 3 is 15. And if we simplify what's in the expression right over here-- once again, it's commutative, so we can swap the order. And it's associative, and so we can swap the groupings. How we group them doesn't matter because it's all multiplication here. This is 2 times 4, which is 6, times x squared times x to the fourth. x squared times x to the fourth is x to the sixth power. And it's all of that to the 1/3 power. And then that is times that-- oh, sorry. Not 6. 2 times 4 is 8. What am I doing? 2 times 4 is 8. x squared times x to the fourth is x to the sixth. I think my brain was adding the exponents and wrote the 6 down. Of course, 2 times 4 is 8, not 6. But we add the exponents. They have the same base. x squared times x to the fourth is x to the sixth. And we're going to raise that to the 1/3 power. And then all of that is times 15. And then we essentially can use this property again. Actually, not that property. We know that this-- well, actually, yeah, exactly this property again. We have something multiplied to a power. This is the exact same thing. This is the exact same thing as 8 to the 1/3 power times x to the sixth to the 1/3 power. And then all of that is being multiplied by 15. And so 8 to the 1/3 power-- that's the same thing as the cube root of 8. You might recognize that 8 is 2 times 2 times 2. So 8 to the 1/3 power is 2. 8 is 2 to the third. So 2 the third to the 1/3 is 2 to the first. 2 times 2 times 2 is 8. And x to the sixth to the 1/3 we know from our exponent properties. That's the same thing as x to the sixth times 1/3 power, or x to the 6 divided by 3 power. Or 6 divided by 3 is 2, or x squared. So that is just x squared. So you have 15 times 2, which gives us 30. So that's these terms right over here. And then you have this term right over here. I want to do that in a different color. And then you have this term right over-- that's not a different color. You have this term right over here is x squared, and you are done. And there's a bunch of ways you could do it. You might not decide to use exponent notation. You could say, look, this is a cube root. I can then take the cube root of the product of both of them. So you don't have to write a 1/3 here. You could just write the cube root of this whole thing over here. And then, depending on how you want to group it and all the rest, you could do this in different orders. As long as you get the right exponent properties, you should get to this same answer.