Exponents and Radicals

The concept of exponentiation, which represents "repeated multiplication" for integer exponents, can be extended to the real numbers. The following laws for exponents apply to all real numbers (and thus to all integers):

(1) x^mxⁿ = x^m+n

(2)
x^m = x^m-n
xⁿ

(3) (x^m)^m = x^mn

(4) (xy)^m = x^my^m

(5) (a/b)^m = a^m/b^m

(6) x⁰ = 1

(7) b^-n = 1/bⁿ

 

EX. x² · x⁴ = x²⁺⁴ = x⁶

 

[(a^-2)(b³)]² = [(a^-2)²(b³)²] = (a^-4)(b⁶)

(a/b)³ = a³ / b³

b^-2 = 1/b²

Exponentiation involving fractional exponents gives rise to the concept of roots and radicals. The relationship between exponents and roots is expressed as follows:

In the above equation, b is known as the radicand, n is the index, and the expression on the right-hand side is the nth root of b. The 2nd root of a number is known as its square root, while its 3rd root is known as its cube root.

EX. 25^1/2 = ±5

If we set a to be the nth root of a real number b, then

a = b^1/n

aⁿ = b^(1/n)n = b¹ = b

The roots of real numbers may be either real or complex numbers. In particular, the nth root of a negative radicand, where n is even, has to be a complex number, since the nth power of any real number, where n is even, has to be a positive number. We will restrict our discussion of exponents and roots to real number solutions.

Given that n is an even integer,

(-b)ⁿ = (-1·b)ⁿ = (-1)ⁿ bⁿ = bⁿ

Therefore, if the nth root of a real number, where n is even, can assume a value a, it can also assume the value -a. To avoid confusion, we define the principal nth root of a real number, where the nth root is a real number, to be the positive nth root of the number. When an algebraic expression addresses the nth root of a number, and the root is a real number, it is often customary to refer to the principal (positive) nth root of that number.

EX. Since 4 ² = (-4)² = 16, the square root of 16 can have the value 4 or -4. Its positive square root is 4. That is,

16^1/2 = 4

By applying the laws of exponentiation, we get the following equality:

By letting m = n

b^m/m b¹ = b

The nth root of a product of two numbers is evaluated as

EX.

Similarly, the nth root of a quotient of two numbers is evaluated as

EX.