Roots

If a square has a side of length 7 feet, then it has an area of 7², or 49 square feet. If a square has an area of 36 square feet, then the length of its side is 6 feet. If we know the length of a side, then we square it to find the area. If we know the area, then we must undo the process of squaring to find the length of a side. Undoing the process of squaring is called taking the square root.

Because 3² = 9, 2³ = 8, and (-2)⁴ = 16, we say that 3 is a square root of 9, 2 is the cube root of 8, and -2 is a fourth root of 16. In general, undoing an nth power is referred to as taking an nth root.

nth Roots

The number b is an nth root of a if bⁿ = a.

Both 3 and -3 are square roots of 9 because 3² = 9 and (-3)² = 9. Because 2⁴ = 16 and (-2)⁴ = 16, there are two real fourth roots of 16: 2 and -2. If n is a positive even integer and a is any positive real number, then there are two real nth roots of a. We call these roots even roots. The positive even root of a positive number is called the principal root. The principal square root of 9 is 3, and the principal fourth root of 16 is 2. When n is even, the exponent 1/n is used to indicate the principal nth root. The principal nth root can also be indicated by the radical symbol .

Exponent 1/n When n Is Even

If n is a positive even integer and a is a positive real number, then a^1/n denotes the positive real nth root of a and is called the principal nth root of a.

Caution

An exponent of n indicates nth power, an exponent of -n indicates the reciprocal of the nth power, and an exponent of 1/n indicates nth root.

We will see later that choosing 1/n to indicate nth root fits in nicely with the rules of exponents that we have already studied.

Example 1

Finding even roots

Evaluate each expression.

a) 4^1/2

b) 16^1/4

c) -81^1/4

d)

Solution

a) Because 2² = 4, we have 4^1/2 = 2. Note that 4^1/2 ≠ -2.

b) Because 2⁴ = 16, we have 16^1/4 = 2. Note that 16^1/4 ≠ -2.

c) Following the accepted order of operations from Chapter 1, we find the root first and then take the opposite of it. Because 3⁴ = 81, we have 81^1/4 = 3 and -81^1/4 = -3.

d) Because , we have .

Note that 2³ = 8 but (-2)³ = -8. The cube root of 8 is 2, and the cube root of -8 is -2. If n is a positive odd integer and a is any real number, then there is only one real nth root of a. We call this root an odd root.

Exponent 1/n When n Is Odd

If n is a positive odd integer and a is any real number, then a^1/n denotes the real nth root of a.

Example 2

Finding odd roots

Evaluate each expression.

a) 8^1/3

b) (-27)^1/3

c) -32^1/5

Solution

a) Because 2³ = 8, we have 8^1/3 = 2.

b) Because (-3)³ = -27, we have (-27)^1/3 = -3.

c) Because 2⁵ = 32, we have -32^1/5 = -2.

We do not allow 0 as the base when we use negative exponents because division by zero is undefined. However, positive powers of zero are defined, and so are roots of zero; for example, 0⁴ = 0, and so 0^1/4 = 0.

nth Root of Zero

If n is a positive integer, then 0^1/n = 0.

Caution

An expression such as (-9)^1/2 is not included in the definition of roots because there is no real number whose square is -9. The definition of roots does not include an even root of any negative number because no even power of a real number is negative.

The expression 3^1/2 represents the unique positive real number whose square is 3. Because there is no rational number that has a square equal to 3, the number 3^1/2 is an irrational number. If we use a calculator, we find that 3^1/2 is approximately equal to the rational number 1.732. Because the square root of 3 is not a rational number, the simplest representation for the exact value of the square root of 3 is 3^1/2.