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Properties of Roots
Properties of Roots. Fundamentally, roots are a special case of exponents. So some of the properties of exponents are the same as the property of roots. In a couple lessons from this one, we’ll make clear the link between roots and exponents. That’s something we’ll discuss in a little more depth.
Right now, we’ll just focus on a couple of big properties of roots. In all the following examples, I’m going to use square roots and assume that the numbers P and Q are positive. In fact, all these properties work for all higher order roots as well, and for the odd roots, the numbers don’t even have to be positive. So, I didn’t wanna clutter up the notation with nth roots everywhere, but just so you know, everything I’m talking about with square roots here, this actually extends to higher order roots.
It’s just rare that you would have to use that on the test. Remember that exponents distribute over multiplication and division. Roots also distribute over multiplication and division. So the root of a product is the product of the roots and the root of a quotient is the quotient of the roots. The multiplication property makes some simplifications much easier.
Suppose we’re told that we had to simplify square root of 12 times square root of 27. Hm. Well, one way to go about this is we could multiply them together. And of course, we’re not just gonna multiply out and get that horrible three digit number, again, we’re gonna think about this. We’re gonna factor out a 3 from that 27 because that will leave us with the perfect square 9.
And factor of 3, I’ll multiply by the 12. Well when I do that, I get another perfect square. So I now have the square of two perfect squares. Separate them out, take the square root of each and it’s 6 times 3 which is 18. So it must be that whatever 12 times 27 is, that must be 18 squared, but fortunately, we didn’t have to figure out that number in order to perform this simplification.
We’ll explore this much more in the next lesson. The division property makes it easier to simplify the square roots of some fractions. So for example, if we had the square root of 4 over 49, we can just take the square root of each of those numbers individually and that equals two sevenths. If we had the square root of 4 over 50, of course, we can simplify a bit first.
We can cancel a factor of 2, then take the square root in the numerator and the denominator. In the numerator, we can’t do anything else. Square root of 2, we’re stuck with that. But in the denominator, we can simplify that to 5. So these are two fractions that get a little bit simplified.
And we’re gonna discuss this process in much more detail a couple of lessons from now. This is, this will be a problem of what happens when you get a square root in the denominator. That didn’t happen here, but it is a problem and we’ll talk about it in a few lessons.
As with exponents, there is also a very tempting mistake to make here. Roots do distribute over multiplication and division. Roots do not distribute over addition and subtraction. So it’s very tempting to say that square root of P plus square root of Q, to set that equal to the square root of P plus Q or the same with subtraction, it’s a very tempting mistake to make, but it is simply not true.
We will see a very important application of the multiplication rule in the next lesson. In summary, roots distribute over multiplication and division. Roots do not distribute over addition and subtraction.
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