# 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?
 If a is negative: If a is zero: If a is positive: an is negative; an is zero; an is positive.

How many real solutions are there to xn = b?
 If b is negative: If b is zero: If b is positive: there's one real solution,which is negative; there's one real solution,which is zero; there's one real solution,which is positive.

Define nb to be this solution, called the real nth root of b (or the real 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.
(We usually call 3b the real cube root of b.)

What is the sign of nb?
 If b is negative: If b is zero: If 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?
 If a is negative: If a is zero: If a is positive: an is positive; an is zero; an is positive.

How many real solutions are there to xn = b?
 If b is negative: If b is zero: If b is positive: there's no real solution; there's one real solution,which is zero. there's two real solutions, one negative and one positive.

Define nb to be the non-negative solution, called the principal nth root of b (or the principal root of b of index n), 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.
(We usually write 2b as simply √b and call it the principal square root of b.)

What is the sign of nb?
 If b is negative: If b is zero: If b is positive: n√b is undefined (or imaginary); n√b is zero; n√b is positive.

## Examples

Find 38.
8 = 23, so 38 = 2.
Find 3−27.
−27 = (−3)3, so 3−27 = −3.
Find √16, meaning 216.
16 = 42 and 4 ≥ 0, so √16 = 4.
Find 4−81.
−81 is negative and 4 is even, so 4−81 is undefined (or imaginary).
Find −38.
38 = 2, so −38 = −2.
Find −3−27.
3−27 = −3, so −3−27 = 3.
Find −√16.
16 = 4, so −√16 = −4.
Find −4−81.
4−81 is undefined (or imaginary), so −4−81 is also undefined (or imaginary).
Find 3x3.
3x3 = x.
Find 3x6.
x6 = (−x2)3, so 3x6 = −x2.
Find √x2.
x2 = (x)2 and x2 = (−x)2; either way, x2 = |x|2 and |x| ≥ 0, so √x2 = |x|.
Find 416x8y4.
16x8y4 = (2x2|y|)4 and 2x2|y| ≥ 0, so 416x8y4 = 2x2|y|.

## 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 = nbm.
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 undefined (or 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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