The exponent of a number says how numerous times to usage the number in a multiplication.

You are watching: X 1 2 times x 2

In this example: 82 = 8 × 8 = 64


So, as soon as in doubt, just remember to compose down every the letters (as plenty of as the exponent tells you to) and see if you can make sense of it.

All you require to recognize ...

The "Laws the Exponents" (also dubbed "Rules that Exponents") come native three ideas:

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The exponent says how many times to usage the number in a multiplication.
*
A negative exponent means divide, because the the contrary of multiplying is dividing
*
A spring exponent choose 1/n way to take the nth root: x(1n) = n√x

If you understand those, climate you understand exponents!

And all the laws listed below are based upon those ideas.

Laws of Exponents

Here are the regulations (explanations follow):


Law instance
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:
xm/n = n√xm =(n√x )m x2/3 = 3√x2 =(3√x )2

Laws Explained

The very first three laws above (x1 = x, x0 = 1 and also x-1 = 1/x) space just part of the organic sequence of exponents. Have a look at this:


Example: powers of 5
.. Etc..
*
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 the table because that a when ... Notification that positive, zero or negative exponents room really component of the same pattern, i.e. 5 times larger (or 5 times smaller) depending upon whether the exponent gets larger (or smaller).

The law that xmxn = xm+n

With xmxn, how many times perform we end up multiplying "x"? Answer: very first "m" times, climate by another "n" times, because that a complete of "m+n" times.


Example: x2x3 = (xx)(xxx) = xxxxx = x5

So, x2x3 = x(2+3) = x5


The regulation that xm/xn = xm-n

Like the previous example, how plenty of times carry out we end up multiplying "x"? Answer: "m" times, climate reduce that by "n" time (because we space dividing), because that a full of "m-n" times.


Example: x4/x2 = (xxxx) / (xx) = xx = x2

So, x4/x2 = x(4-2) = x2


(Remember the x/x = 1, so every time you see an x "above the line" and also one "below the line" you can cancel castle out.)

This regulation can likewise show you why x0=1 :


Example: x2/x2 = x2-2 = x0 =1


The legislation that (xm)n = xmn

First you main point "m" times. Then you have to carry out that "n" times, because that a full of m×n times.


Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12

So (x3)4 = x3×4 = x12


The regulation that (xy)n = xnyn

To show how this one works, just think that 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 regulation that xm/n = n√xm =(n√x )m

OK, this one is a little more complicated!

I indicate you check out Fractional index number first, so this makes much more sense.

Anyway, the vital idea is that:

x1/n = The n-th source of x

And so a fractional exponent like 43/2 is yes, really saying to execute a cube (3) and a square root (1/2), in any kind of order.

Just remember native fractions that m/n = m × (1/n):


Example: x(mn) = x(m × 1n) = (xm)1/n = n√xm


The bespeak does not matter, for this reason it additionally works because that m/n = (1/n) × m:


Example: x(mn) = x(1n × m) = (x1/n)m = (n√x )m


Exponents of index number ...

What around this example?

432

We do the exponent in ~ the top first, so us calculate that 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 discover it hard to remember all these rules, then remember this:

you have the right to work lock out as soon as you recognize thethree ideas near the height of this page:

The exponent claims how many times to use the number in a multiplicationA negative exponent means divideA spring exponent choose 1/n way to take the nth root: x(1n) = n√x

Oh, One more Thing ... What if x = 0?

Positive Exponent (n>0) 0n = 0
Negative Exponent (n-n is undefined (because splitting by 0 is undefined)
Exponent = 0 00 ... Ummm ... See below!


The Strange instance of 00

There room different arguments for the correct value of 00

00 can be 1, or maybe 0, so some world say the is really "indeterminate":

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x0 = 1, so ... 00 = 1
0n = 0, therefore ...

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00 = 0
When in doubt ... 00 = "indeterminate"

323, 2215, 2306, 324, 2216, 2307, 371, 2217, 2308, 2309
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