Using itWe could use the nth root in a question like this: Question: , what is "n"? Answer: 5 × 5 × 5 × 5 = 625, so n=4 (ie 5 is used 4 times in a multiplication) Or we could use "n" because we want to say general things: Example: If n is odd then Why "Root" ... ?
PropertiesNow we know what an nth root is, let us look at some properties: Multiplication and DivisionYou can "pull apart" multiplications under the root sign like this:
This can help you simplify equations in algebra, and also make some calculations easier: Example: It also works for division:
Example: Addition and SubtractionBut you cannot do that kind of thing for additions or subtractions ! It is an easy trap to fall into, so beware. It also means that, unfortunately, additions and subtractions can be hard to deal with when under a root sign.
Exponents vs RootsAn exponent on one side of the "=" can be turned into a root on the other side of the "=":
nth root of a-to-the-nth-powerWhen a value has an exponent of n and you then take the nth root, you get the value back again (or sometimes the absolute value of it):
nth root of a-to-the-mth-powerNow let's see what happens when the exponent and root are different values (m and n).
So ... you can move the exponent "out from under" the nth root, which may sometimes be helpful. But there is an even more powerful method ... you can combine the exponent and root to make a new exponent, like this:
That is because the nth root is the same as an exponent of (1/n):
You might like to read about Fractional Exponents next to find out why! |