Exponents and Radicals

First we shall consider exponents. . . Exponents Let n ∈ N , a ∈ R .

Then a n = a · a · · · a ( n times). If m is a negative integer and the real number a = , then a m = a − m . Note that for any a = , we have a a = a − = a = . It is also easy to see the following properties.

Properties of Exponents (i) For m, n ∈ Z and a = , we have a m a n = a m + n . (ii) For m, n ∈ Z and a = , we have a m a n = a m − n . . .

Radicals Question: For a = and r ∈ Q , is it possible to define a r ? First let us consider the case when r = n , n ∈ N . Suppose there is a real number y ∈ R such that y = a n . Then we must have y n = a .

This problem is basically to finding inverse function of y = x n . In order to understand better let us consider the graphs of the following functions: (i) f ( x ) = x n , n ∈ N (ii) g ( x ) = x n + , n ∈ N – – f ( x ) = x g ( x ) = x h ( x ) = x – – – g ( x ) = x g ( x ) = x g ( x ) = x – – – Basic Algebra From these two figures it is clear that the function g : R → R given by g ( x ) = x n + , n ∈ N is one-to-one and onto and hence its inverse function from R onto R exists. But f : R → [ , ∞ ) given by f ( x ) = x n , n ∈ N is onto but not one-to-one. However, f is one-to-one and onto if we restrict its domain to [ , ∞ ) .

This is helpful in understanding n th root of a real number. So we have two cases; Case When n is even. In this case y n = a is not meaningful when a < . So no such y exists when a < .

Assume that a > . If y is a solution to x n = a, then − y is also solution to x n = a . Case When n is odd. In this case no such problem arises as in Case .

For y ∈ R , there is a unique x ∈ R such that y = x n . Based on the above observation we define radicals as follow. Definition . (i) For n ∈ N , n even, and b > , there is a unique a > such that a n = b .

(ii) For n ∈ N , n odd, b ∈ R , there is a unique a ∈ R such that a n = b . In both cases a is called the n th root of b or radical and is denoted by b /n or n √ b (i) If n = , then n th root is called the square root; if n = , then it is called cube root. (ii) Observe that the equation x = a , has two solutions x = a, x = − a ; but a = | a | . (iii) Properties of exponents given above are still valid for radicals provided each of the individual terms are defined.

(iv) Note that for n ∈ N and a = we have ( a n ) /n = | a | if n is even , a if n is odd . For example, p ( − ) = / = , / = and ( − ) = − . For any rational r = m n , m ∈ Z , n ∈ N , with gcd ( m, n ) = and for a > we define a r = a m n = ( a /n ) m . For example, / = ( / ) = = .

But ( − ) / has no meaning in real number system because there is no real number x such that x = ( − ) . . . Exponential Function Observe that for any a > and x ∈ R , a x can be defined.

If a = , we define x = . So we shall consider a x , x ∈ R for < a = . Here a x is called exponential function with base a . Note that a x may not be defined if a < and x = m for even m ∈ N .

This is why we restrict to a > . Also, a x > for all x ∈ R . It does also satisfy the following: Properties of Exponential Function For a, b > and a = = b (i) a x + y = a x a y for all x, y ∈ R , (ii) a x a y = a x − y for all x, y ∈ R , . Exponents and Radicals (iii) ( a x ) y = a xy for all x, y ∈ R , (iv) ( ab ) x = a x b x for all x ∈ R , (v) a x = if and only if x = .

. Let us consider f ( x ) = a x , x ∈ R where a = . Now f ( x ) = x , x ∈ R . Let us show that f is one-to-one and onto.

Suppose f ( u ) = f ( v ) for some u, v ∈ R . Then, we have u = v , which implies that u v = , ⇒ u − v = . So, u − v = and hence u = v . Thus f is a one-to-one function.

– – . – Figure . : f ( x ) = x From the graph it is clear that values of f ( x ) = x increase as x values increase and the range of f is ( , ∞ ) . So as = , we have x > for all x > and x < for all x < .

Observe that f : R → ( , ∞ ) is onto. . Let us consider a = . Let g ( x ) = x = x , x ∈ R .

From the graph it is clear that the values of g ( x ) = x decrease as x values increase and g ( R ) = ( , ∞ ) . Also, g ( ) = we have g ( x ) > for all x < and g ( x ) < for all x > . Remark: Exactly same arguments as above would show that an exponential function f ( x ) = a x , for any base < a = , is one-to-one and onto with domain R and codomain ( , ∞ ) .