The exponent of a number states how many times to use the number in a multiplication.

You are watching: E^x times e^x

In this example: 82 = 8 × 8 = 64


So, once in doubt, simply remember to create down every the letters (as numerous as the exponent tells you to) and see if you can make sense of it.

All you need to recognize ...

The "Laws that Exponents" (also called "Rules the Exponents") come indigenous three ideas:

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The exponent says how numerous times to usage the number in a multiplication.
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A negative exponent means divide, due to the fact that the opposite of multiplying is dividing
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A spring exponent favor 1/n method to take the nth root: x(1n) = n√x

If you know those, climate you know exponents!

And every the laws listed below are based on those ideas.

Laws of Exponents

Here are the legislations (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 around Fractional Exponents:
xm/n = n√xm =(n√x )m x2/3 = 3√x2 =(3√x )2

Laws Explained

The an initial three laws over (x1 = x, x0 = 1 and x-1 = 1/x) room just part of the natural sequence that exponents. Have a look in ~ this:


Example: strength of 5
.. Etc..
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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 for a if ... Notification that positive, zero or an unfavorable exponents are really component of the same pattern, i.e. 5 times bigger (or 5 times smaller) depending upon whether the exponent gets bigger (or smaller).

The law that xmxn = xm+n

With xmxn, how countless times perform we finish up multiply "x"? Answer: an initial "m" times, climate by another "n" times, for a complete of "m+n" times.


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

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


The legislation that xm/xn = xm-n

Like the previous example, how many times perform we finish up multiply "x"? Answer: "m" times, climate reduce that through "n" time (because we room dividing), for 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, therefore every time you watch an x "above the line" and one "below the line" you deserve to cancel lock out.)

This regulation can likewise show girlfriend why x0=1 :


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


The legislation that (xm)n = xmn

First you multiply "m" times. Then you have to perform that "n" times, for a complete 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 present how this one works, simply think that re-arranging all the "x"s and "y"s together in this example:


Example: (xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3


The regulation that (x/y)n = xn/yn

Similar to the vault example, just re-arrange the "x"s and also "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 an ext complicated!

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

Anyway, the important idea is that:

x1/n = The n-th source of x

And for this reason a fractional exponent prefer 43/2 is really saying to execute a cube (3) and a square root (1/2), in any type of order.

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


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


The order does no matter, for this reason it also works for m/n = (1/n) × m:


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


Exponents of exponents ...

What around this example?

432

We perform the exponent in ~ the top first, so us calculate the 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 tough to remember all these rules, then remember this:

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

The exponent says how plenty of times to usage the number in a multiplicationA negative exponent means divideA fractional exponent choose 1/n means to take it the nth root: x(1n) = n√x

Oh, One an ext 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 case of 00

There space different debates for the correct worth of 00

00 might be 1, or probably 0, for this reason some people say that is yes, really "indeterminate":

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

See more: Can You Heat Up Mayo Nnaise? Is It Safe To Microwave Mayonnaise On A Sandwich

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

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