
Algebra PreAlgebra and Basic Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 25th, 2016, 09:32 PM  #1 
Member Joined: Mar 2015 From: Los Angeles Posts: 73 Thanks: 7  what bases and exponents are defined on real numbers
I'm interested in knowing what bases and exponents have a meaning/definition in real numbers. Tell me if any of this is right. I think that $0^0$ is undefined. $ 0^a = 0 $ but only for $ a > 0 $. $b^0 = 1$ for all real $b$ except $b=0$. Is this right so far? Then we get into negative bases and rational exponents. Suppose we have $$ b \in \mathbb{R}, b <0, m \in \mathbb{N}, n \in \mathbb{N} $$ Then $ b^{m \over n} $ has a real solution only if $n$ is odd. Correct? It has imaginary or complex solutions if $n$ is even. Also, what about this? For $ b \in \mathbb{R}, b < 0, a \in \mathbb{R}$, then $b^a$ generally does not have a solution unless the above conditions are met. But does it have complex solutions? Like, does $ (1)^\pi $ have any complex solutions? 
March 26th, 2016, 05:33 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2587 Math Focus: Mainly analysis and algebra 
$(1)^\pi=\left(\mathrm e^{\mathrm i \pi}\right)^\pi=\mathrm e^{\mathrm i \pi^2}$ which is a complex number of unit modulus and argument $\pi^2$. Once you get into negative bases, the construction may exist in the real numbers or not as you point out, but we don't usually deal with functions having negative bases because they are not connected (smoothly) connected. 
March 26th, 2016, 07:23 AM  #3  
Member Joined: Mar 2015 From: Los Angeles Posts: 73 Thanks: 7  Quote:
$(1) = e^{ 3 i \pi}$ and $ (1)^\pi = e^ {3 i \pi^2 }$ which is a different number as the angle $3 i \pi^2$ is not coterminal with $ i \pi^2$. Or does this have no practical use?  

Tags 
bases, defined, exponents, numbers, real 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Complex numbers raised to exponents and divided...  Toyboy  Complex Analysis  7  September 12th, 2012 05:04 AM 
Simplify exponents with different bases  daigo  Algebra  3  May 13th, 2012 06:54 PM 
logarithms and exponents of extremely large or small numbers  chanther  Algebra  3  December 29th, 2011 07:03 PM 
exponential functions real number exponents  ChloeG  Algebra  5  November 14th, 2011 01:25 AM 
Real Number Exponents  ChloeG  Algebra  3  May 18th, 2011 12:00 AM 