Laws of Exponent
Laws of Exponents
Here are the Laws (explanations follow): Law | Example | x1 = x | 61 = 6 | x0 = 1 | 70 = 1 | x-1 = 1/x | 4-1 = 1/4 | | | xmxn = xm+n | x2x3 = x2+3 = x5 | xm/xn = xm-n | x6/x2 = x6-2 = x4 | (xm)n = xmn | (x2)3 = x2×3 = x6 | (xy)n = xnyn | (xy)3 = x3y3 | (x/y)n = xn/yn | (x/y)2 = x2 / y2 | x-n = 1/xn | x-3 = 1/x3 | And the law about Fractional Exponents: | | |
Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:
The law that xmxn = xm+n
With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
Example: x2x3 = (xx)(xxx) = xxxxx = x5
So, x2x3 = x(2+3) = x5
The law that xm/xn = xm-n
Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, thenreduce that by "n" times (because you are dividing), for a total of "m-n" times.
Example: x4/x2 = (xxxx) / (xx) = xx = x2
So, x4/x2 = x(4-2) = x2
(Remember that x/x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)
This law can also show you why x0=1 :
Example: x2/x2 = x2-2 = x0 =1
The law that (xm)n = xmn
First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.
Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12
So (x3)4 = x3×4 = x12
The law that (xy)n = xnyn
To show how this one works, just think of re-arranging all the "x"s and "y" as in this example:
Example: (xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3
The law that (x/y)n = xn/yn
Similar to the previous example, just re-arrange the "x"s and "y"s
Example: (x/y)3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x3/y3
The law that
OK, this one is a little more complicated!
I suggest you read Fractional Exponents first, or this may not make sense.
Anyway, the important idea is that: x1/n = The
Here are the Laws (explanations follow): Law | Example | x1 = x | 61 = 6 | x0 = 1 | 70 = 1 | x-1 = 1/x | 4-1 = 1/4 | | | xmxn = xm+n | x2x3 = x2+3 = x5 | xm/xn = xm-n | x6/x2 = x6-2 = x4 | (xm)n = xmn | (x2)3 = x2×3 = x6 | (xy)n = xnyn | (xy)3 = x3y3 | (x/y)n = xn/yn | (x/y)2 = x2 / y2 | x-n = 1/xn | x-3 = 1/x3 | And the law about Fractional Exponents: | | |
Laws Explained
The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this:
The law that xmxn = xm+n
With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.
Example: x2x3 = (xx)(xxx) = xxxxx = x5
So, x2x3 = x(2+3) = x5
The law that xm/xn = xm-n
Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, thenreduce that by "n" times (because you are dividing), for a total of "m-n" times.
Example: x4/x2 = (xxxx) / (xx) = xx = x2
So, x4/x2 = x(4-2) = x2
(Remember that x/x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)
This law can also show you why x0=1 :
Example: x2/x2 = x2-2 = x0 =1
The law that (xm)n = xmn
First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.
Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12
So (x3)4 = x3×4 = x12
The law that (xy)n = xnyn
To show how this one works, just think of re-arranging all the "x"s and "y" as in this example:
Example: (xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3
The law that (x/y)n = xn/yn
Similar to the previous example, just re-arrange the "x"s and "y"s
Example: (x/y)3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x3/y3
The law that
OK, this one is a little more complicated!
I suggest you read Fractional Exponents first, or this may not make sense.
Anyway, the important idea is that: x1/n = The