Indices and Standard Form

Slide 1Chapter

10

It’s 144 000 000 km,

or 1.440 00 x 108 km.

or 1.440 00 x 100 000 000 km,

Earth to Moon: 384 835 km

Earth to Sun: 144 000 000 km

We know that the Sun is very far from the Earth. But how far exactly is it? How many digits are needed to represent this distance?

What about the distance from the Earth to the Moon?

Indices and Standard Form

Slide 2Chapter

10

Moon:

7.3 x 1019 metric tons

Sun:

22 x 1027 metric tons

Earth:

6.6 x 1021 metric tons

Besides the distances mentioned earlier, even the masses of the Earth, the Moon and the Sun are pretty large too.

The numbers involved are so large that we make use of INDICES to represent them.

Indices and Standard Form

Slide 3Chapter

10

Diameter of a human hair: 0.000 025 4 m

We can rewrite it as

25.4 x 0.000 001 m,

or 25.4 x 10-6 m

Besides representing very large numbers, we can also make use of INDICES to represent very small numbers.

Some examples: diameter of a strand of hair, size of an atom, size of a bacterium

What are INDICES then?

Indices and Standard Form

Slide 4Chapter

10

1.440 00 x 108 km 25.4 x 10-6 m

These numbers are called INDICES.

We make use of INDICES to represent extremely

LARGE or small numbers.

INDICES saves us from writing long string of

digits, saving time and effort, and reducing the

chance of missing out digits.

Indices and Standard Form

Slide 5Chapter

10

22 x 1027

= 22 000 000 000 000 000 000 000 000

Imagine having to write out 22 x 1027 which is the value of the mass of the Sun, in full in your assignment about the Solar System.

Indices and Standard Form

Slide 6Chapter

10

23

2 x 2 x 2 can be written as 23,

where Index / Exponent

Base

Indices and Standard Form

Slide 7Chapter

10

a5a3 x aa x a x a x aam = x aa x a … a

m times

x aa x a

Indices and Standard Form

Slide 8Chapter

10Below are the laws of indices for expressions with a common base.

amx an = am+n

am an = am-n

(am)n = am n

Indices and Standard Form

Slide 9Chapter

10

Summary

amx an = am+n

am an = am–n

(am)n = am n = amn

Indices and Standard Form

Slide 10Chapter

10

a3

=x

(ab3

(a (a b) b) b)x=

(a) (a) (a)

(a b)3

=

(b) (b) (b)

Below are the laws of indices for expressions with a common index.

Indices and Standard Form

Slide 11Chapter

10

=(a) (a) (a)

(b) (b) (b)

a3

b3

=

a

b

a

b

a

b

a

b

a

b

a

b

3

Indices and Standard Form

Slide 12Chapter

10Summary

amx bm = (ab)m

am bm = ( )ab

m

Indices and Standard Form

Slide 13Chapter

10

2-3

a0 = 10

a-n =

=n√a2

2√4

3√82

18123

1an

a1n41 281 3

Zero Index

Negative Index

Fractional Index

Indices and Standard Form

Slide 14Chapter

10

=n√a2

3√8a

1n81 3

a n =n√a

4

1 mm

8 3 =3√

1 22648

More on Fractional Index

Indices and Standard Form

Slide 15Chapter

10Summary

a0 = 1

a-n =1

an

= n√aa

1n =

n√ama

mn

Indices and Standard Form

Slide 16Chapter

10

4x 16= 42

2x

Equations Involving Indices

=

Indices and Standard Form

Slide 17Chapter

10

1.440 00 x 108

km

Index(plural: indices)Earth to Sun:

Then what is this form of expressing numbers known as?

1.440 00 x 108

A, where 1 ≤ A < 10 n is an integer.In general, A x 10n

— standard form of 144 000 000

Indices and Standard Form

Slide 18Chapter

10

Standard form for very large numbers

Yes! So, in standard form:

1.440 00 x 108

144 000 000 Numbers larger than 1,

move to the leftIs 1 ≤ A < 10 now?

38 6

Indices and Standard Form

Slide 19Chapter

10Standard form for very small numbers

Yes! So, in standard form:1.440 00 x 10-7

0.000 000 144

Numbers smaller than 1,move to the right.

Is 1 ≤ A < 10 now?

73 6