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Simplifying square-root expressions

CCSS.Math:

Video transcript

let's get some practice simplifying radical expressions that involve variables so let's say I have two times the square root of 7x times 3 times the square root of 14 x squared pause the video and see if you can simplify taking any perfect squares out of multiplying and then taking any perfect squares out of the radical sign well let's first just multiply to this thing so we can change the order of multiplication this is going to be the same thing as 2 times 3 times the square root of 7 x times the square root of 14 x squared and so this is going to be equal to 6 times and then the product of two radicals you could view that as the square root of the product so 6 times the square root of and I'll actually I'll just leave it like this 7 times X and then let me actually factor 1414 is 2 times 7 times x squared and actually let me let me extend my radical sign a little bit all right and the reason why I didn't multiply it out obviously we could have multiplied in our head x times x squared is X to the third and we could have said all right 7 times 14 is what 98 we could have done that but when you're trying to factor out perfect squares it's actually easier if it's in this factored form here especially because from a variable point of view you could view this as a perfect square already and then 14 is not a perfect square 7 isn't a perfect square but 7 times 7 is 7 times 7 is a perfect square that is 49 of course so let's rewrite this a little bit to see what we can do this is going to be 6 times and I could write it like this the square root of and let's put all the perfect squares first so 7 times 7 that is 49 that's those two x-squared 49 x squared and then I could once again separate the two radicals right over here so whatever else is left so I've already used the 707 the x squared I have a 2x left times 2x hopefully appreciate that these two things are equivalent I could have put one big radical sign over 49 x squared times 2x which should have been exactly what you have there but if you're taking the radical of the product of things that's the same thing as the product of the radicals this comes straight out of our exponent properties but what's valuable about this is we now see this is 6 times now we could take the square root of 49 x squared that's going to be 7 X square root of 49 is 7 square root of x squared is going to be X and then we multiply that times the square root of 2x times the square root of 2x and so now we're in the homestretch 6 times 7 is 42 x times the square root of 2x and the key thing to appreciate is I keep using this property that a radical of products or the square root of products is the same thing as the product of the square roots so even this step that I did here if you wanted you could have had an intermediary step you could have said that the square root of 49 x squared is the same thing as square root of 49 times the square root of x squared which would get us square root of 49 is 7 square root of x squared is X right over there let's do let's do another one of these so let's say I have square root of 2a times the square root of 14 a to the third times the square root of 5a so like always pause this video and see if you can simplify this on your own multiply them and then take all the perfect squares out of the radical so let's multiply first so this is going to be the same thing as the square root of let's see 2 times 14 times five so let me actually just I'm just going to two and five our prime 14 I can factor it as two times seven so this is going to be two times instead of fourteen I'm going to write two times seven and then times five and then we have a times a to the third times a well actually let me just write that as a to the fifth we have a to the first times e to the third times e to the first add X once you get a to the fifth now what perfect squares do we have here well we already see a perfect square in terms of two times two and then a to the fifth isn't a perfect square if you think in terms of the variable a but you could view that as a perfect square a to the fourth times a so let's rearrange this a little bit and so this is going to be equal to the square root of let me put my perfect squares out front the square root of four two times two times a to the fourth and then let me put mine on perfect squares times I have a seven a five and an A that I haven't used yet so seven times five is thirty-five so it's 35 a and now just like we said before we could let me do it we could say hey look this is the same thing as the square root of four times the square root of a to the fourth just using exponent properties and then times the square root of 35 a now principal root of four is positive two you could view this as a positive square root and then square root of a to the fourth the principal root is going to be a squared and then we're going to have that times the square root of 35 a and we're done let's do one more example and this time we're going to involve two variables which as you'll see isn't that much more complicated so let's simplify the square root of 72 X to the third Z to the third so the keya can we factor 72 is not a perfect square but kit is there a perfect square someplace in there and you immediately see that if you try to factor it you get it you get thirty-six times two and 36 of course is a perfect square and likewise X to the third and D two-thirds are not each perfect squares but they each have an x squared and Z squared in them so let me rewrite this this is the same thing this is the same thing as I can write well we put all my perfect squares out front so I have 36 36 I'm gonna take an x squared out x squared I'm going to take a Z squared out Z squared and then what we're left with is we took a 36 out so we're left with a 2 times 2 and if we took an x squared out of this we're left with just an X X to the third divided by x squared is X to X and then Z to the third divided by Z squared is just Z and you can verify this multiply this all out you should be getting you should be getting exactly what we have here I do that aligned on the Z to differentiate so it doesn't look like my twos 36 times 2 is 72 x squared times X is X to the 3rd Z squared times Z is Z to the third but now this is pretty straightforward to factor because let's let me just I'll do more steps and you would probably do if you're doing it on your own but that's because the whole point here is to learn so 2x Z so that's just using exponent properties and so everything here is a perfect square this is going to be the square root of 36 times the square root of x squared times the square root of Z squared which is going to be square root of 36 is principal root of 36 is 6 principal root of x squared is X principal root of Z squared is Z and we're going to multiply that times square root of 2 X Z and we are done