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RootsIf a square has a side of length 7 feet, then it has an area of 7^{2}, 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 3^{2} = 9, 2^{3} = 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 b^{n} = a.
Both 3 and -3 are square roots of 9 because 3^{2} = 9 and (-3)^{2} = 9. Because 2^{4} = 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 2^{2} = 4, we have 4^{1/2} = 2. Note that 4^{1/2} ≠ -2. b) Because 2^{4} = 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 3^{4} = 81, we have 81^{1/4} = 3 and -81^{1/4} = -3. d) Because , we have . Note that 2^{3} = 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 2^{3} = 8, we have 8^{1/3} = 2. b) Because (-3)^{3} = -27, we have (-27)^{1/3} = -3. c) Because 2^{5} = 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, 0^{4} = 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}. |