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 ⁿ√b 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:

ⁿ√b = a;
aⁿ = b.

(We usually call ³√b the real cube root of b.)

What is the sign of ⁿ√b?

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 are two real solutions, one negative and one positive.

Define ⁿ√b 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:

ⁿ√b = a;
aⁿ = b, and a ≥ 0.

(We usually write ²√b as simply √b and call it the principal square root of b.)

What is the sign of ⁿ√b?

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.

We will look at imaginary numbers later on. These allow us to make sense of ⁿ√b when b is negative and n is even, although we will only consider the case when n = 2 in this course.

Fractional exponents

Because it makes most of the rules of exponents continue to work, we define b^1/n to mean ⁿ√b. We can generalize this to any rational number m/n in lowest terms:

b^m/n = ⁿ√b^m.

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 (not even imaginary, but completely 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.)

When b is positive, it's possible to define b^x (as another positive number) even when x is irrational, but we won't pursue that in this course. (If b is negative and x is irrational, then b^x is imaginary. If b is zero and x is irrational and positive, then b^x is also zero. If b is zero and x is irrational and negative, then b^x is completely undefined.)

Examples

Find ³√8.: 8 = 2³, so ³√8 = 2.
Find ³√−27.: −27 = (−3)³, so ³√−27 = −3.
Find √16, meaning ²√16.: 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 −³√8.: ³√8 = 2, so −³√8 = −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 ³√x³.: ³√x³ = x.
Find ³√−x⁶.: −x⁶ = (−x²)³, so ³√−x⁶ = −x².
Find √x².: x² = (x)² and x² = (−x)²; either way, x² = |x|² and |x| ≥ 0, so √x² = |x|.
Find ⁴√16x⁸y⁴.: 16x⁸y⁴ = (2x²|y|)⁴ and 2x²|y| ≥ 0, so ⁴√16x⁸y⁴ = 2x²|y|.

^* 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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