EXPONENTS - BASICS, SOLVED EXAMPLES AND EXERCISE. LINKS TO FURTHER STUDY

Please study  Basic Algebra before Exponents,
if you have not already done so.

There, we introduced the idea of literal number.

We also explained the basic operations like addition,
subtraction, multiplication and division on literal numbers.

Here, we deal with special cases of multiplication with literal numbers.

When a (literal) number is multiplied more than once,
we use a special notation to represent it.

We explain this special notation with examples.

We explain the terms : Base, Index, Exponential form, Expanded form.

We provide solved examples and problems for practice
with answers, to help to understand the conepts covered.

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Exponents : Introduction

we have seen x x y is written as x.y or simply xy.

As per this notation,
x x x (ie x multiplied twice) is to be written as x.x or xx.

similarly x multiplied three times is to be written as xxx.
x multiplied four times is to be written as xxxx.

As the literal number is multiplied more and more number of times, we have to repeat writing the literal number more and more number of times.

Is it not tedious?

We have a solution.

When the same literal number is multiplied more than once,
instead of writing the same literal number again and again,
a special notation is used.

The notation is,
writing a number (which tells how many times the literal number is multiplied), to the right (and slightly above) of the literal number being multiplied.

Let us see some examples:

(i) x . x is written as x², which means x is multiplied twice.
This is read as x power 2 or x square.

(ii) y . y . y is written as y³, which means y is multiplied thrice.
This is read as y power 3 or y cube.

(iii) a . a . a . a is written as a⁴, which means a is multiplied four times. This is read as a power 4.

(iv) 6⁵ means 6x6x6x6x6. Read as 6 power 5.
Here the number being multiplied is not literal number but a numerical.

(v) 9x⁶ means 9. x . x . x . x . x . x . Read as "nine ' x ' power 6".

(vi) 3a³b² means 3 a . a . a . b . b . Read as "three ' a ' cube ' b ' square.

(vii) ( x⁄y )² means ( x⁄y ).( x⁄y ). Read as ' x by y whole square'.

The number which is being multiplied with itself is called BASE. The number written to the right (and slightly above) of the base is called an EXPONENT or INDEX.

For Example: In x⁵, x is called the base, 5 is the index.
In (4ab )³, 4ab is called the base. 3 is the index.

Index 2 is called 'square' instead of power 2.
Index 3 is called 'cube' instead of power 3.
Indices 4,5,6,7, ....etc are called power 4,5,6,7, .... etc respectively.

What about Index '1'?
Here is the answer.

When a number has no index, the index is understood to be one. Thus x and x ¹ are the same.

x ⁰ = 1 where x is any base (not zero).

What about Index '-1'?
For the time being, you may note

x ^-1 = 1⁄ x where x is any base (not zero).

Basics of Exponents and Powers

Based on the discussion above, we have

First power of x = x¹ = x.
Second power of x = x² = x x x ( i.e. x multiplied twice).
Third power of x = x³ = x x x x x ( i.e. x multiplied thrice).
Fourth power of x = x⁴ = x x x x x x x ( i.e. x multiplied four times).
....................... and so on
nth power of x = xⁿ = x x x x x.......n times ( i.e. x multiplied n times).

Here x is any real number and n is any positive integer.

In xⁿ, x is called the BASE and n is called the EXPONENT.
xⁿ is called the EXPONENTIAL FORM.
x.x.x.x.....n times is called the EXPANDED FORM.

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