# Crash Course, Mathematics: Exponents

## What are exponents?

Exponents are very important part of mathematics at every level. To understand them lets start with an example. We know that the expression 2 x 2 x 2 can also be written as 3. This is an exponential representation implying that 2 has to be multiplied by itself 3 times. Here number 2 is the base while number 3 is the exponent.

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## Generalization:

an = a x a x a x . . .  n times

Here a is the base while n is the exponent

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## Rules of Exponents:

### Rule 0:       a0 = 1          [any number to the power 0 is 1]

One of the most important rules in Mathematics. It is always true for any number except for a = 0.

Example:   90 = 1;        1000 = 1;    100000 = 1;     [anything except 0]0= 1

### Rule 1:

am x an = am + n

Proof:        am = a x a x a . . . m times

an = a x a x a . . . n times

am x an = a x a x a . . . [m times] x a x a x a . . . [n times]

am x an = a x a x a . . .  m + n times

am x an = am + n              . . . Hence Proved

Example:   24 x 23 = [2 x 2 x 2 x 2] x [2 x 2 x 2] = 27 = 24 + 3

### Rule 2:

am ÷ an = am – n

Proof:        am = a x a x a . . . m times

an = a x a x a . . . n times

$$\frac { { a }^{ m } }{ { a }^{ n } } =\frac { a\quad *\quad a\quad *\quad a\quad *\quad .\quad .\quad .\quad m\quad times }{ a\quad *\quad a\quad *\quad a\quad *\quad .\quad .\quad .\quad n\quad times }$$

=> $$\frac { { a }^{ m } }{ { a }^{ n } }$$ = a * a * a . . . m – n times

am ÷ an = am – n              . . . Hence Proved

Example:   35 ÷ 33 = 3 x 3 x 3 x 3 x 3 / 3 x 3 x 3 = 3 x 3 = 32 = 35 – 3

### Rule 3:

a-m = 1/am

Proof:        a-m = a0 – m

= a0 / am                       . . . From Rule 2

= 1/am                           . . . From Rule 1

. . . Hence Proved

Example:   3-2 = 1/32 = 1/9

A very common occurrence: 1/x = x-1

### Rule 4:

(ab)n = an x bn               &                (a/b)n = an / bn

Proof:        (ab)n = ab x ab x ab . . . n times

= a x a x a . . . n times x b x b x b . . . n times

= an x bn               . . . Hence Proved

Example:   (15 x 14)3 = 153 x 143 = 3375 x 2744 = 9261000

Check: (15 x 14)3 = 2103 = 210 x 210 x 210 = 9261000

. . . Q.E.D.

### Rule 5:

(am)n = am x n

Proof:        (am)n = (a x a x a . . . m times)n

= (an x an x an . . .  m times)            . . . Rule 4

= (an + n + n . . .  m times)                    . . . Rule 2

= (an x m)                                            . . . Hence Proved

Example:   (33)2 = 33 x 2 = 36 = 729

Check: (33)2 = (27)2 = 729              . . . Q.E.D.

Rules of exponents are extremely important to remember. One must not confuse one with another which is often a mistake done by beginners. Suggestion is to keep a formula sheet always with you and learn whenever you get a chance.

Important Note: These formulas give out wrong results when base is negative or the numbers are complex, so be careful

I will be publishing on different topics at least once a week. These will incorporate formulas and important rules to follow while attempting Math questions.

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