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November 11th, 2017, 05:36 AM   #1
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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.
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November 11th, 2017, 05:54 AM   #2
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$$-4\frac1a=-4\frac{1}{a^1}=-4a^{-1}$$
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November 11th, 2017, 06:18 AM   #3
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Quote:
Originally Posted by greg1313 View Post
$$-4\frac1a=-4\frac{1}{a^1}=-4a^{-1}$$
Thank you so much, I see now! So -4/a is the same thing as -4*1/a. That's using the multiplicative inverse property. If you are dividing x by a number, it is the same thing as multiplying x by the reciprocal of that number!
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November 11th, 2017, 06:37 PM   #4
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Quote:
Originally Posted by Force3DSpace View Post
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.
Yes that's a very good question. Exponents are often presented as a list of magic rules like $x^{-1} = \frac{1}{x}$ but what does that mean?

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   ...
Then one day someone asks, What if we read this right to left, and just see what happens if we continue the pattern?

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
So when we say that $2^0 = 1$, that's not something you need to make sense of at first. You can just think of it as saying that $0$ is above $1$ when we start with a positive power of 2 and keep subtracting 1 on top and dividing by 2 below.

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.
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Last edited by Maschke; November 11th, 2017 at 07:08 PM.
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November 12th, 2017, 04:56 AM   #5
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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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