# Indices

### Summary

MUST Remember all these Rules of Indices:

1.   ${ a }^{ m }\quad \times \quad { a }^{ n }\quad =\quad { a }^{ m+n }$

2.   $\quad \frac { { a }^{ m } }{ { a }^{ n } } \quad =\quad { a }^{ m-n }$

3.   $\quad ({ { a }^{ m }) }^{ n }\quad =\quad { a }^{ m\times n }$

4.   $\quad { a }^{ m }\quad \times \quad { b }^{ m }\quad =\quad { (a\quad \times\quad b)\quad }^{ m }$

5.   $\quad { a }^{ m }\quad \div \quad { b }^{ m }\quad =\quad { (\frac { a }{ b } )\quad }^{ m }$

6.   $\quad { a }^{ 0 }\quad =\quad 1$

7.   $\quad { a }^{ -n }\quad =\quad \frac { 1 }{ { a }^{ n } }$

8.

### What are Indices?

Just like other operations in maths (+, -,  x,), Indices are a type of math operations. However they don’t have a fix symbol for it.

We can identify it from the way a certain number is written e.g  $\quad { a }^{ n }$ where n is called an exponent/index/power and a is called the base. They tell us to take a number and multiply it by itself a certain no. of times which is called repeated multiplication.

It is defined as:

• When n is a positive integer,  $\quad { a }^{ n }$  is defined as:
$\quad { a }^{ n }\quad =\quad a\quad \times \quad a\quad \times \quad a\times \quad a\quad .............\quad { a }_{ n }$

### Some basic rules of Indices that you must remember are:

#### Rule #1:

When we multiply two numbers which have the same base, we add their powers.

${ a }^{ m }\quad \times \quad { a }^{ n }\quad =\quad { a }^{ m+n }$

#### Rule #2:

When we divide two number which have the same base, we subtract their powers.

$\quad \frac { { a }^{ m } }{ { a }^{ n } } \quad =\quad { a }^{ m-n }$

#### Rule #3:

When we have a power to the power of something else, we multiply the powers together.

$\quad ({ { a }^{ m }) }^{ n }\quad =\quad { a }^{ m\times n }$

To be able to understand these rules better, let’s go through some examples.

#### Examples

Q. Solve:           ${ 2 }^{ 4 }\quad \times \quad { 2 }^{ 3 }$

Solution:

We know Rule #1 will apply here, hence for:

${ 2 }^{ 4 }\quad \times \quad { 2 }^{ 3 }$

$=\quad { 2 }^{ 4\quad +\quad 3 }$

Ans:            $=\quad { 2 }^{ 7 }\quad =\quad 128$

Q.         ${ x }^{ 5 }\quad \times \quad { x }^{ 10 }\quad \times { \quad x }^{ 4 }$

Solution:

Apply Rule #1:

${ =\quad x }^{ 5 }\quad \times \quad { x }^{ 10 }\quad \times { \quad x }^{ 4 }$

${ =\quad x }^{ (5\quad +\quad 10\quad +\quad 4) }$

Ans:            ${ =\quad x }^{ 19 }$

Q.         ${ (x+y) }^{ 3 }{ (x+y) }^{ 5 }$

Solution:

$=\quad { (x+y) }^{ 3 }{ (x+y) }^{ 5 }$

Ans:            $=\quad { (x+y) }^{ 8 }$

Q.         $\frac { { x }^{ \frac { 3 }{ 4 } } }{ x }$

Solution:

Apply Rule #2:

$=\quad \frac { { x }^{ \frac { 3 }{ 4 } } }{ x }$

$=\quad { x }^{ \frac { 3 }{ 4 } \quad -\quad 1 }$

Ans:            $=\quad { x }^{ -\frac { 1 }{ 4 } }$

Q.         $\frac { { x }^{ 3 }{ y }^{ 2 } }{ x{ y }^{ 3 } }$

Solution:

Apply Rule #2:

$=\quad \frac { { x }^{ 3 }{ y }^{ 2 } }{ x{ y }^{ 3 } }$

$=\quad { x }^{ 3\quad -\quad 1 }{ y }^{ 2\quad -\quad 3 }$

Ans:            $=\quad { x }^{ 2 }{ y }^{ -1 }$

Q.         ${ (a{ b }^{ 2 }) }^{ 4 }$

Solution:

Apply Rule #3:

$=\quad { (a{ b }^{ 2 }) }^{ 4 }$

$=\quad { a }^{ 4 }{ ({ b }^{ 2 }) }^{ 4 }$

Ans:            $=\quad { a }^{ 4 }{ b }^{ 8 }$

Furthermore, there are some more important rules which we should be familiar. We should understand how and when they are applied as all these rules are used a lot of times when integrating, differentiating and in other places.

#### Rule #4:

When we multiply two numbers which have different base but same powers, we then multiply the base numbers but the power remains the same.

$\quad { a }^{ m }\quad \times \quad { b }^{ m }\quad =\quad { (a\quad \times\quad b)\quad }^{ m }$

#### Rule #5:

When we divide two numbers which have different base but same powers, we then divide the base numbers but the power remains the same.

$\quad { a }^{ m }\quad \div \quad { b }^{ m }\quad =\quad { (\frac { a }{ b } )\quad }^{ m }$

#### Rule #6:

When power of any constant or variable is 0 then that constant or variable is equal to 1.

$\quad { a }^{ 0 }\quad =\quad 1$

#### Rule #7:

A negative power means the constant or variable is on the bottom line of the fraction.

$\quad { a }^{ -n }\quad =\quad \frac { 1 }{ { a }^{ n } }$

#### Rule #8:

Roots can be written as powers as well.

#### Examples

Q. Solve:           ${ 3 }^{ 2x\quad +\quad 1 }\quad +\quad 26({ 3 }^{ x })\quad -\quad 9\quad =\quad 0$

Solution:

Let:

${ 3 }^{ x }\quad =\quad y\quad \rightarrow \quad$        equation 1

Then:

$3{ (3) }^{ 2x }\quad +\quad 26({ 3 }^{ x })\quad -\quad 9\quad =\quad 0$

$3y^{ 2 }\quad +\quad 26y\quad -\quad 9\quad =\quad 0$

$3y^{ 2 }\quad +\quad 27y\quad -\quad y\quad -\quad 9\quad =\quad 0$

$3y(y\quad +\quad 9)\quad -\quad 1(y\quad +\quad 9)\quad =\quad 0$

$(y\quad +\quad 9)(3y\quad -\quad 1)\quad =\quad 0$

$y\quad =\quad \frac { 1 }{ 3 } \quad and\quad y\quad =\quad -9$

As $y\quad =\quad -9$ is not possible, we take $y\quad =\quad \frac { 1 }{ 3 }$

Put y in equation 1:

${ 3 }^{ x }\quad =\quad y$

${ 3 }^{ x }\quad =\quad \frac { 1 }{ 3 }$

${ 3 }^{ x }\quad =\quad { (3) }^{ -1 }$

This shows that:

Ans:            $x\quad =\quad -1$

#### Reference

1. CGP AS Level Mathematics Edexcel