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 2­³. 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.

Generalization:

aⁿ = a x a x a x . . .  n times

Here a is the base while n is the exponent

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:

a^m x aⁿ = a^{m + n}

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

aⁿ = a x a x a . . . n times

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

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

a^m x aⁿ = a^{m + n}              . . . Hence Proved

Example:   2⁴ x 2³ = [2 x 2 x 2 x 2] x [2 x 2 x 2] = 2⁷ = 2^{4 + 3}

Rule 2:

a^m ÷ aⁿ = a^{m – n}

Proof:        a^m = 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

a^m ÷ aⁿ = a^{m – n}              . . . Hence Proved

Example:   3⁵ ÷ 3³ = 3 x 3 x 3 x 3 x 3 / 3 x 3 x 3 = 3 x 3 = 3² = 3^{5 – 3}

Rule 3:

a^-m = 1/a^m

Proof:        a^-m = a^{0 – m}

= a⁰ / a^m                       . . . From Rule 2

= 1/a^m                           . . . 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:      

(a^m)ⁿ = a^{m x n}

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

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

= (a^{n + n + n . . .  m times})                    . . . Rule 2

= (a^{n x m})                                            . . . Hence Proved

Example:   (3³)² = 3^{3 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.

If you wish to discuss further on “Exponents” or want to clarify any Math related query, go to:

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