Roots

Just as subtraction reverses addition and division reverses multiplication, so taking roots reverses raising to powers. By considering how raising to the power of a natural number affects whether a number is positive or negative, we can see under what conditions and how much this operation may be reversed.

Raising to the power of an odd number

What is the sign of an when n is an odd natural number?
a is negative a is zero a is positive
an is negative an is zero an is positive

How many real solutions are there to xn = b?
b is negative b is zero b is positive
one real solution, negative one real solution, zero one real solution, positive

Define nb to be this solution, called the nth root of b (or the root of b of index n). In other words, these two statements mean the same thing when n is an odd natural number: What is the sign of nb?
b is negative b is zero b is positive
nb is negative nb is zero nb is positive

Raising to the power of an even number

What is the sign of an when n is an even natural number?
a is negative a is zero a is positive
an is positive an is zero an is positive

How many real solutions are there to xn = b?
b is negative b is zero b is positive
no real solution one real solution, zero two real solutions, one negative and one positive

Define nb to be the non-negative solution, called the principal nth root of b, if such a solution exists. In other words, these two statements mean the same thing when n is an even natural number: What is the sign of nb?
b is negative b is zero b is positive
nb is imaginary nb is zero nb is positive

Fractional exponents

Because it makes most of the rules of exponents continue to work, we define b1/n to mean nb. We can generalize this to any rational number m/n in lowest terms:
bm/n = n(bm).
If b is positive, then this always exists (and is positive). If b is zero, then this is zero if m is positive and undefined if m is negative.* (Since m/n is in lowest terms, n must be positive.) If b is negative, then this is negative if m and n are both odd, positive if m is even and n is odd, and imaginary if m is odd and n is even. (Since m/n is in lowest terms, m and n cannot both be even.)
* In the special case where b and m are both zero, modern mathematics defines 00 = 1. However, our textbook takes the old-fashioned view that 00 is undefined. To avoid confusion, I will never test you on 00.
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