**Notes**
**Ncert Solutions**
**Assignments**
## What is exponent and Base?

This can be explained with the below example

We know

2

^{2} = 2×2

3

^{3} = 3×3×3

So it is known how we interact with positive exponent

## Negative exponents

Here we will be looking how to interact with these type of expression

2^{-10}

11^{-1}

Here we can see that exponents are negative

Now negative exponents can be converted into positive exponents like this

2

^{-10} = 1/2

^{10}
11

^{-1}= 1/11

So

we can say that for any non-zero integer

*a*,

a

^{-m} = 1/a

^{m}
where

*m *is a positive integer.

*a*^{m}* *is the multiplicative inverse of

*a*^{-m}
**Example**

a) 2^{-3}

b) 3^{-2}

**Solution**

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

a) 2^{-3} = ½^{3} = 1/8

b) 3^{-2} = 1/3^{2} =1/9

## How to express the decimal number in exponent form

We already know that any non -decimal number can be expressed in exponent form like below

1215 =1000+ 200 +10 +5 = 1 × 10

^{3} + 2 × 10

^{2} + 2 × 10

^{1} + 5 × 10

^{0}
So these are all positive exponents

Decimal number can be expressed as exponents also

1215.15 =1000+ 2-00 +10 +5 +.10+.05= 1 × 10

^{3} + 2 × 10

^{2} + 2 × 10

^{1} + 5 × 10

^{0} +1/10 + 5/100

=1 × 10

^{3} + 2 × 10

^{2} + 2 × 10

^{1} + 5 × 10

^{0} +10

^{-1} + 5 ×10

^{-2}
So this has both the negative and positive exponents.

## Laws of Exponents

Here are the laws of exponents when

*a *and

*b *are non-zero integers and

*m*,

*n *are any integers.

a

^{-m} = 1/a

^{m}
a

^{m} / a

^{n} = a

^{m-n}
(a

^{m} )

^{n} = a

^{mn}
a

^{m} x b

^{m } = (ab)

^{m}
a

^{m} / b

^{m } = (a/b)

^{m}
a

^{0} =1

(a/b)

^{-m} =(b/a)

^{m}
(1)

^{n} = 1 for infinitely many

*n*.

(-1)

^{p} =1 for any even integer p

These laws can be used to solve the exponents problems

**Example:**

(-3)^{3} × (2/3)^{3}

**Solution**

(-3)^{3} × (2/3)^{3}

=(-1)^{3} × (3)^{3 } × (2)^{3} × (3)^{-3} as a^{m} x b^{m } = (ab)^{m} and a^{m} / b^{m } = (a/b)^{m}

= (-1)^{3} × (2)^{3}

=-8

**Important Note**

*a*^{n} = 1 only if *n *= 0. This will work for any *a *except *a *= 1 or *a *= –1. For *a *= 1, 1^{1} = 1^{2} =1^{3} = 1 or (1)^{n} = 1 for infinitely many *n*.

For *a *= –1,(–1)^{0} = (–1)^{2} = (–1)^{4} = (–1)^{-2} = ... = 1 or (–1)^{p} = 1 for any even integer *p*.

**Watch this tutorial for more explanation About exponents and power**

## Use of Exponents to Express Small Numbers in Standard Form

In previous classes we have learnt how to convert large number into exponents form. We can express small number in the standard exponent form also

Let’s take an example to explain

.00001

**Step 1:** Count of Number of places after decimal

Here it is 5

**Step 2:** On the basis of counting in step 1,the number can be written as

.00001 = 1/100000=1/10^{5 } = 1X 10^{-5}

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

Basically the decimal has been moved to the place after digit

Another example would be

.00019

**Step 1:** In these cases, we count of Number of places after decimal to the first digit only

Here it is 4

**Step 2:** On the basis of counting in step 1,the number can be written as

.0001 = 1.9/10000=1.9/10^{4 } = 1.9X 10^{-4}

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

**Watch this tutorial for more explanation About Use of Exponents to Express Small Numbers in Standard Form**
**Practice Questions**

a) .00156

b) .005402

c) .1234

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