Introduction to Powers and Roots

In an expression of the form x^a, the name given to a is the index, or power, or exponent. For any number x and a positive integer n, we define xⁿ in terms of multiplication by:

xⁿ = x × x × x × x × ... × x (where there are n factors)

i.e. xⁿ is the product of n x's.

For example

x² = x × x

x⁵ = x × x × x × x × x

x³ = x(x²)

For x != 0 we can define negative integer powers by the reciprocal of xⁿ, that is

x^-n = 1 / xⁿ

For example

x^-2 = 1 / x²

x^-5 = 1 / x⁵

Finally, for x != 0 , we define

x⁰ = 1

so that we have defined x^a for any integer value of a.

What about fractional powers?

First we define the n-th roots of any number x member of R .

If two non-zero numbers x and y are related by:

Then y is called an n-th root of x and we write:

i.e.

y = x^(1/n) is equivalent to y^n = x (n is a member of N)

Examples

2 = 8^1/3	because	2³ = 8
3 = 9^1/2	because	3² = 9
-3 = 9^1/2	because	(-3)² = 9
	because

Remarks

1. For every positive number x, there is a unique positive n-th root for all n member of N . The notation nth root of x is used to denote this unique positive n-th root (but note that: x^(1/n) will almost always be interpreted as this unique positive n-th root).

In particular, for x > 0, root x always means the unique positive square root of x; the other (negative) square root is - root x .

e.g.

2. For every positive number x, there are two n-th roots for all even n member of N

e.g.

16^(1/4) = ±4; (4 = 4th root 16, -4 = -[4th root 16])

But again, in calculation, 16^1/4 will almost always be interpreted as the positive value +4.

3. For every negative number x < 0, there is a unique negative n-th root if n is odd and no real n-th root if n is even.

e.g.

8^1/3 = 2

(-4)^1/3 does not exist in the set of real numbers R.

4. Note the last comment "in the set of real numbers". When you study complex numbers you will discover that the situation regarding n-th roots is much simpler! In the complex number system every non-zero number has n distinct n-th roots for all positive integers n.

Now we can define the fractional powers of a number, x^a, where a is a fraction. Firstly, for any fraction p/q written in its lowest terms: p is any integer q is a member of N , q is any positive integer p is a member of Z and p and q have no common factors (this last point is important):

y = x^p/q is equivalent to y^q = x^p y = (x^p)^1/q

y^p/q = (x^p)^1/q = (x^1/q)^p

The expression on the right is often easier to calculate, especially if qth root x is an integer.

Whether x^p/q exists at all, or is unique, or is two-valued, depends on x, p and q. Remember, p and q should have no common factors and in particular should not both be even.

Examples

8^2/3 = (8²)^1/3 = 64^1/3 = 4

8^2/3 = (8^1/3)² = 2² = 4

(-32)^-3/5 = ((-32)^1/5)^-3 = (-2)^-3 = -1/8

16^3/4 = (16^1/4)³ = (±2)³ = ±8

(-8)^4/3 = ((-8)^1/3)⁴ = (-2)⁴ = 16

(-16)^3/4 = ((-16)^1/4)³ does not exist.

N.B. For x > 0, x^p/q is almost always interpreted as the positive value

even when strictly it is two-valued (i.e. when p is odd and q is even). That is, in Example 3 above, +8 will almost always be taken as the value of 16^3/4.

The fact that p and q have no common factors is particularly important to ensure, for example, that

(-8)^1/3 = -2

is properly defined as -2. Otherwise we could have:

-2 = (-8)^1/3 = (-8)^2/6 = ((-8)²)^1/6 = 64^1/6 = ±2

and often 64^1/6 will be interpreted as +2. This erroneous answer is introduced by writing the power 1/3 as 2/6.

Thus we have defined x^a, where a can be any rational number, that is a number that can be written as a ratio p/q. Extending this to irrational powers such as x^(root 2) is intuitively reasonable - your calculator will give you the answer - but it is theoretically difficult. Remember, irrational numbers cannot be written in the form of a ratio, so try to decide for yourself what really means.