Indices & Standard Form F2 2013

8/11/2019 Indices & Standard Form F2 2013

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Form 2 [CHAPTER 3: INDICES & STANDARD FORM]

1 www.smcmaths.webs.com| C. Camenzuli

Chapter 3: Indices & Standard Form

Indices

3.1 Index laws for Multiplication and Division

Multiplication

25 23 = (2222 2) (2 2 2)

= 2 2 2 2 2 2 2 2

= 28

Therefore we can say that:2

523= 25+3= 28

In general,xm

xn=x

m+n(add the indices)

Division

3533 = 3 3 3 3 33 3 3

= 3 3 3 3 33 3 3

/ / /

/ / /= 32

Therefore we can say that

3533= 35 3= 32

In general,xm

xn=x

m-n(subtract the indices)

Example 1: Write each of these with a single index number:

(i) 25 24

(ii) 59 55


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(iii) 4 4

(iv) 77 72

(v) x7 x6

(vi) y4 y4

Example 3: Write each of these with a single index number.

(i) 65 66 610

(ii) 75 7 73

(iii) 57 53 59

(iv) 62 64 6

Example 4: Find n :a) 3n 35 = 32

4 = 41Go to fullsizeimage


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b) 52x 5

n= 5

10

c) 54= 5 x 5n

3.2 Index laws for bracketsWe can often meet expressions of the form (23)2or (22)3

These can be solved with the multiplication law.

(23)2= 23 23= 23+3= 26

(22)3= 22 22 22= 22+2+2= 26

In general (am

)n= a

mn

Example 1:Write each of the these with a single index number:

(i) (72)4

(ii) (84)4

(iii) (33)3

(iv) (x2)5

(v) (y8)2


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3.3 Index zero

Powers of zero

Any number or letter to the power of zero is equal to 1.

40= 150= 180= 1999990= 1x

0= 1y0= 1

So for all non-zero values of x, x0

= 1

3.4 Negative indices and fractions with indicesNegative powers

2

2

12

2-

=

In general,1nn

aa

-=

Example1: Work out the value of the following:

(i) 2-1

(ii) 3-2

(iii) 10-3

(iv) ( )2


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(v) 2

(vi) ( )4

(vii) ( )-1

(viii) ( )-3

(ix) ( )-1

(x) ( )-2

(xi) ( )-1

Summary of index laws:

The 5 index laws

1)xmxn=xm+n(add the indices)

2)xmxn= xm n(subtract the indices)

3) for all non-zero values of x,x0= 1

4)

5) (xm)n=xmn


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Standard Form

3.5 Ordinary Numbers to Standard FormIt often occurs that we need to work out calculations with very large numbers. It

could be a problem to write these numbers out so we must try and find an

alternative method.

The age of the earth is 4.6 thousand millionyears.

If we write this out in numbers we get:

4 600 000 000 years

This large number can be written in a way called standard formto make it

simpler.

When we write a number in standard form it is in the formax 10n

where1a < 10. This means that ahas to be a number larger than 1 and strictly

smaller than 10 and n is an integer.

9000 can be written as 9 1000

To be in standard form it has to be written as 9 103

9 is the number between 1 and 10, that is a = 9 and n = 3

Example 1: Write 571 in standard form

amust be a number between 1 and 10

a= 5.71

The point has moved 2 digits to the left. Therefore n = 2

Therefore: 571 = 5.71 102


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Example 2:Write 250 000 in standard form




Example 6: Write 0.0089 in standard form

amust be a number between 1 and 10

a= 8.9

The point has moved 3 digits to the right. Therefore n = -3

Therefore: 0.0089 = 8.9 10-3





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3.6 Standard Form to ordinary numbers

Example 1: Write 7.3 10 as an ordinary numberMove the point 4 times to the right (number must get bigger)

Example 2:Write 6.54 10-2 as an ordinary number

Move the point 2 times to the left (number must get smaller)

Example 3:Write 7.98654 10-1as an ordinary number

Example 4: Write 5.61655 106as an ordinary number

Using the calculator

With a scientific calculator, standard form can be easily worked out.

For this method we use the EXP or 10xbutton

Example

5.6 104

1)Digit in the number (5.6)

2)Press the EXP button

3)Digit the power of 10

4)Press the =

Indices & Standard Form F2 2013

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