Square Roots
Introduction & Simplification:-
"Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3:
The " " symbol is called the "radical"symbol. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) The expression "
Simplifying Square-Root Terms:-
To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:
Note that the value of the simplified radical is positive. While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. When you solve the equation x2 = 4, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression
Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:
- Simplify
· There are various ways I can approach this simplification. One would be by factoring and then taking two different square roots:
The square root of 144 is 12
- Simplify
Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?
- Simplify
Multiplying Square Roots:-
The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.
Simplifying multiplied radicals is pretty simple. We use the fact that the product of two radicals is the same as the radical of the product, and vice versa.
- Write as the product of two radicals:
Okay, so that manipulation wasn't very useful. But working in the other direction can be helpful:
- Simplify by writing with no more than one radical:
Adding (and Subtracting) Square Roots:-
Simplify:
Simplify:
·
Conjugates / Dividing by Square Roots:-
· Simplify
I do the multiplication:
Then I complete the calculations by simplifying:
Dividing by Square Roots
Just as you can swap between the multiplication of radicals and a radical containing a multiplication, so also you can swap between the division of roots and one root containing a division.
- Simplify:
I can simplify this by working inside, and then taking the square root:
...or else by splitting the division into two radicals, simplifying, and cancelling:
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· Prepared by:-
