# 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:
• nb = a;
• an = b.
What is the sign of nb?
 b is negative b is zero b is positive n√b is negative n√b is zero n√b 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:
• nb = a;
• an = b, and a ≥ 0.
What is the sign of nb?
 b is negative b is zero b is positive n√b is imaginary n√b is zero n√b 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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