Calculating with root expressions

Do you know what a square root and a cube root are? We are now going to go deeper into the roots and learn how to work with square roots and other root expressions. If you multiply two square roots it's the same as multiplying the radicands under the same root sign. The root of 5 times the root of 20 is the same as the root of 100. And this is useful since 100 is a perfect square, and you can calculate the result without rounding.

While 5 and 20 are not perfect squares. If you would have calculated each root separately a rounding error would be unavoidable. ​Division of square root works the same way. The root of 12 is divided by the root of 3, is the same as the root of 12 divided by 3, which is the root of 4, which is 2. These rules apply to all roots, not only square roots, as long as they are of the same degree. In other words they have the same number above the root sign.

Adding and subtracting roots is not this simple. You first have to calculate the value, rounded off if needed, and then add or subtract. The root of 9 plus the root of 16 does not equal to the root of 25. The sign for not equal to looks like a crossed over equal sign. This rules applies to cube roots and roots of the fourth, fifth, and other degrees as well.

Sometimes it can be practical to factor an expression under the root to solve a problem. 54 is not a perfect square, so the root gives an irrational number that has to be rounded. You can simplify this root by factoring it using the bigger perfect square that is a factor of the radicand. ​Nine is a perfect square, and it is a factor of fifty-four. So, we factor the radicand to nine times six. Now you can use your knowledge about root multiplication, and split the root into two factors.

The root of nine and the root of six. The root of nine is exactly three, so you can simplify it and get three times the root of six. The root of fifty-four is equal to three times the root of six. You have taken a whole root expression and rewritten it in a simplified form. A root that cannot be factored using a perfect square is already in simplified form.

Like this one, the root of 13. It is not divisible by four or nine, which are perfect squares smaller than thirteen therefore, it cannot be written in a simplified form without rounding. Let's go over the rules again. Multiplication of roots of the same degree is the same as multiplying the radicands under the same root sign. In the same way you can perform a division under a root sign or between two roots.

Addition and subtraction on the other hand, do not follow the same rule. Here, you need to calculate each root individually, and add or subtract the results. Afterwards it get a little more complicated. Roots whose radicand can be factored using a perfect square can be simplified. If the number you are taking the root of is divisible by four, nine, sixteen, twenty-five or any other perfect squares, you can factor it and calculate a root that does not need to be rounded.