**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 **2 ^{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**.

## What else you can do inside qs leap ?

**Generalization****:**

a^{n} = a x a x a x . . . n times

Here **a is the base** while **n is the exponent**

## What else you can do inside qs leap ?

**Rules of Exponents:**

**Rule 0: **a^{0} = 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: **9^{0} = 1; 100^{0} = 1; 10000^{0} = 1; **[anything except 0] ^{0}**= 1

**Rule 1:**

a^{m} x a^{n} = a^{m + n}

**Proof:** a^{m} = a x a x a . . . m times

a^{n} = a x a x a . . . n times

a^{m} x a^{n} = a x a x a . . . [m times] x a x a x a . . . [n times]

a^{m} x a^{n} = a x a x a . . . m + n times

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

**Example: **2^{4} x 2^{3} = [2 x 2 x 2 x 2] x [2 x 2 x 2] = 2^{7} = 2^{4 + 3}

**Rule 2:**

a^{m} ÷ a^{n} = a^{m – n}

**Proof: **a^{m} = a x a x a . . . m times

a^{n} = a x a x a . . . n times

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

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

**Example: **3^{5} ÷ 3^{3} = 3 x 3 x 3 x 3 x 3 / 3 x 3 x 3 = 3 x 3 = 3^{2} = 3^{5 – 3}

**Rule 3:**

a^{-m} = 1/a^{m}

**Proof: **a^{-m} = a^{0 – m}

= a^{0} / a^{m} . . . From Rule 2

= 1/a^{m} . . . From Rule 1

. . . Hence Proved

**Example: **3^{-2} = 1/3^{2} = 1/9

**A very common occurrence: **1/x = x^{-1}

**Rule 4: **

(ab)^{n} = a^{n} x b^{n} & (a/b)^{n} = a^{n} / b^{n}

**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

= a^{n} x b^{n} . . . Hence Proved

**Example:** (15 x 14)^{3} = 15^{3} x 14^{3} = 3375 x 2744 = 9261000

Check: (15 x 14)^{3} = 210^{3} = 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}

= (a^{n} x a^{n} x a^{n} . . . m times) . . . Rule 4

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

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

**Example: **(3^{3})^{2} = 3^{3 x 2} = 3^{6} = 729

Check: (3^{3})^{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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