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November 11th, 2017, 04:36 AM  #1 
Newbie Joined: Nov 2017 From: Florida Posts: 4 Thanks: 1  Help needed to understand properties of negative exponents, with particular problem.
Please, I'd like to know how $\displaystyle 4/a$, became $\displaystyle 4*a^1$. I understand a little about negative exponents, for example for $\displaystyle x^1$ you first find the reciprocal of the base, and then raise that to the positive exponent. $\displaystyle 1/x^1$. Also, I read that when you have fractions raised to a negative power, you can flip that fraction and raise it to the positive power, but I still don't understand how the division became multiplication.

November 11th, 2017, 04:54 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,950 Thanks: 1141 Math Focus: Elementary mathematics and beyond 
$$4\frac1a=4\frac{1}{a^1}=4a^{1}$$

November 11th, 2017, 05:18 AM  #3 
Newbie Joined: Nov 2017 From: Florida Posts: 4 Thanks: 1  
November 11th, 2017, 05:37 PM  #4  
Senior Member Joined: Aug 2012 Posts: 2,343 Thanks: 732  Quote:
There's a perfectly sensible explanation for all of the exponent laws. It's easiest to use a specific example, base 2. We know that $2 = 2$, $2 \times 2 = 4$, and so forth. We invent the notation $2^n$ as a shorthand for multiplying $2$ by itself $n$ times. Then we make ourselves a handy little chart. Code: 1 2 3 4 5 6 7 8  ... 2 4 8 16 32 64 128 256 ... As you go right to left from any point, you subtract one from the number on the top; and you divide the bottom number by 2. Continuing this pattern, we get Code: 4 3 2 1 0 1 2 3 4 5 6  ... 1/16 1/8 1/4 1/2 1 2 4 8 16 32 64 Likewise the rule $x^{n} = \frac{1}{x^n}$ is just a shorthand way of expressing the left hand part of the chart. It just tells you how to read off the top/bottom numbers. In other words you don't need to try to figure out what this "means," but you don't need to take it as completely arbitrary rule. There's an underlying pattern. Eventually it all becomes second nature. If you're doing exponents now you'll see exponentials and logarithms soon. The key is that going top to bottom in our chart is the exponential function base 2. Going bottom to top is the base 2 logarithm. I should mention that conceptually we fill in all the real numbers as well as the integer powers of 2. The real numbers are on the top row; and all the positive real numbers are on the bottom. [You can't take the log of a number that's negative or zero]. You can visualize the exponent rules too. $2^{n + m} = 2^n 2^m$ for example. That's because addition (and subtraction) above the line corresponds to multiplication and division below the line. Likewise going the other way, $\log_2(xy) = \log_2 x + \log_2 y$. Hope this is helpful. Last edited by Maschke; November 11th, 2017 at 06:08 PM.  
November 12th, 2017, 03:56 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Another way of looking at it: As long as x is a positive integer, it is easy to define a^x as "a multiplied by itself x times". That is $\displaystyle a^2= a*a$, $\displaystyle a^3= a*a*a$, etc. It also easy to see that $\displaystyle (a^x)(a^y)= a^{x+ y}$ ($\displaystyle a^x$ is "x" a's multiplied together, $\displaystyle a^y$ is "y" a's multiplied together. When you combine those, you have x+ y a's.) But what about x^0? If we want that very nice "$\displaystyle (a^x)(a^y)= a^{x+ y}$" to still be true we must have $\displaystyle (a^x)(a^0)= a^{x+ 0}$. But since 0 is the additive identity, $\displaystyle a^{x+ 0}= a^x$. From $\displaystyle (a^x)(a^0)= a^x$, dividing both sides by $\displaystyle a^x$ (requiring that a not be 0) we have $\displaystyle a^0= 1$. And for negative powers, look at $\displaystyle (a^x)(a^{x})$ where x is a positive number. We still want $\displaystyle (a^x)(a^y)= a^{x+ y}$ to be true so we must have $\displaystyle (a^x)(a^{x})= a^{x x}= a^0= 1$. That is, we must have $\displaystyle a^{x}= \frac{1}{a^x}$ (again requiring that a not be 0). 

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