Exponents are a shorthand used for repeated multiplication.
Remember that when you were first introduced to multiplication it was as a shorthand for repeated addition.
For example, you learned that:
Here are some other examples:
If b is a positive real number and n is a positive integer, then there is exactly one positive real solution to
When one speaks of the n-th root of a positive real number b, one usually means the principal n-th root.
where m is an integer and n is a positive integer.
Rational powers m/n, where m/n is in lowest terms, are positive if m is even, negative for negative b if m and n are odd, and can be either sign if b is positive and n is even.
- If n is even, then xn = b has two real solutions;
- if b is positive, which are the positive and negative nth roots.
- The equation has no solution in real numbers if b is negative.
- If n is odd, then xn = b has one real solution.
- The solution is positive if b is positive and negative if b is negative.
Examples of exponents graphed can be seen in this figure .