mathslaws of exponents
Answers
Answer:
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas: The exponent says how many times to use the number in a multiplication. A fractional exponent like 1/n means to take the nth root: If you understand those, then you understand exponents!
Answer:
So an Exponent saves us writing out lots of multiplies!
Example: a7
a7 = a × a × a × a × a × a × a = aaaaaaa
Notice how we wrote the letters together to mean multiply? We will do that a lot here.
Example: x6 = xxxxxx
The Key to the Laws
Writing all the letters down is the key to understanding the Laws
Example: x2x3 = (xx)(xxx) = xxxxx = x5
Which shows that x2x3 = x5, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:
pencil paper The exponent says how many times to use the number in a multiplication.
turn over A negative exponent means divide, because the opposite of multiplying is dividing
pie slice
A fractional exponent like 1/n means to take the nth root: x^(1/n) = n-th root of x
If you understand those, then you understand exponents!
And all the laws below are based on those ideas.
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:
x^(m/n) = n-th root of (x^m) = (n-th root of x)^m x^(2/3) = 3rd root of (x^2) = (3rd root of x)^2
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:
Example: Powers of 5
.. etc.. exponent 5x larger smaller
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04
.. etc..
Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).
The law that xmxn = xm+n
With xmxn, how many times do we 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 do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we 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"s 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 fractional exponents
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 n-th Root of x
And so a fractional exponent like 43/2 is really saying to do a cube (3) and a square root (1/2), in any order.
Just remember from fractions that m/n = m × (1/n):
Example: fractional exponents
The order does not matter, so it also works for m/n = (1/n) × m:
Example: fractional exponents
Exponents of Exponents ...
What about this example?
432
We do the exponent at the top first, so we calculate it this way:
Start with: 432
32 = 3×3: 49
49 = 4×4×4×4×4×4×4×4×4: 262144
And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when you understand the
three ideas near the top of this page
Oh, One More Thing ... What if x = 0?
Positive Exponent (n>0) 0n = 0
Negative Exponent (n<0) 0-n is undefined (because dividing by 0 is undefined)
Exponent = 0 00 ... ummm ... see below!
The Strange Case of 00
There are different arguments for the correct value of 00
00 could be 1, or possibly 0, so some people say it is really "indeterminate":
question mark x0 = 1, so ... 00 = 1
0n = 0, so ... 00 = 0
When in doubt ... 00 = "indeterminate