nth Roots

This brief post is going to explain to you what nth roots are.  Following posts will show you how to work with them in your equations.

Going back to elementary algebra lessons, you were taught the concepts of square roots.  For example, the square root of 25 is 5.  This notion was explained, but likely never generalized much past saying "what times what gives you your number?"  However, like many things in the mathematics world, there is far more to roots than this.

Enter the concepts of "nth roots".

Everyone knows the symbol for square root, and that it means "what times what gives the number underneath this symbol".  Some of you may have seen a tiny 3 written on the top left just above the root sign.  What this means is a touch more complicated: "what times what times what gives the number underneath this symbol?"  It is called the cube root.

It's always simpler to go forwards before going backwards, so let me show you one:

2 x 2 x 2 = 8

You can say 2³ (2 to the power of 3) equals 8.  Straight-forward, right?  Now, go backwards.  Find the cube root of 8.  Well, in this case, "what times what times what" is "2 x 2 x 2", so the cube root of 8 is 2.

So then, we now understand square roots, and cube roots.  Now, naturally, we can expand on those.  In general terms, nth roots can be defined as this:

a^n = b     (a to the power of n equals b)

and therefore a is the nth root of b,

where n is a natural number, and a and b are real numbers.

Some examples should hopefully clear up the complexity of what I just wrote.

3 and -3 are square roots of 9, because 3² = 9

4 and -4 are fourth roots of 64, because 4⁴=64

2 and -2 are sixth roots of 64, because 2⁶=64

Further, what these examples should demonstrate to you is that all EVEN roots of postive numbers occur in pairs, with there being one positive and one negative number for each.  Therefore, to distinguish between the two, there has been accepted a common notation:

The positive, or principal, nth root is designated as the root sign with the nth root number over it:   ⁿ√
E.g. ⁴√16 = 2, and NOT -2
To indicate -2, you write the initial expression as -ⁿ√
E.g. -⁴√16 = -2

On the other hand, all ODD roots only occur singly.  Such as 2 is the fifth root of 32, whereas -2 is the fifth root of -32.  (Write it out to see the differences.  -2 x -2 x -2......)

A little bit more nomenclature, just so you always know what is going on now:
What you have always knows as the "square root sign" is technically called the radical sign.
The number underneath the radical sign is called the radicand, and the number n used to indicate the root is called the index.

To summarize all of this, we can revise our original definition to this:

If n is a natural number, and a and b are non-negative real numbers, then

ⁿ√b = a, if and only if b=aⁿ

^{The number a is the prinicpal nth root of b}

Also:

If a and b are negative and n is an odd natural number, then

ⁿ√b = a, if and only if b=aⁿ

^{So there you have it.  You now know what is meant by nth roots.  In my next post, I will go over some of the properties of nth roots that will make your homework questions a lot easier.}

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