The notation 32 and 23 is known as index form. The small digit is called the index number or power.
We have already seen that 32 = 3·3 = 9, and that 23 = 2·2·2 = 8.

In each case, the index number tells us how many times we should multiply the numbers together.

When the index number is two, we say "squared"
When the index number is three, we say "cubed"
When the index number is grater than three, we say "to the power of"

All scientific calculators have a "power" botton [xy]. This is particulary useful when tyhe index number is large. (After all, you're very likely not to make a mistake if you try to calculate 410 by typing 4·4·4·4·4·4·4·4·4·4 into your calculator!)

 4 shift xy 10 =

INDEX LAWS FOR MULTIPLICATION AND DIVISION

Multiplication

23= 2·2·2
so, 23·25= 2·2·2·2·2·2·2·2= 28
25=2·2·2·2·2

Can you see what happened? We have 3 twos from 23 and 5 twos from 25, so alltogether we had 8 twos.

In general,
2m·2n= 2(m+n)

#### How Do You Multiply Two Numbers With Exponents?

Division

If we divide 25 by 23:
So 25: 23= 22
In general            2m: 2n= 2(m-n)

#### How Do You Divide Two Numbers With Exponents?

SQUARE ROOT

We already know that 3·3 can be written as 32 (3 squared)

The opposite of squaring a number is called finding the square root.

Examples:

The square rootof 16 is 4 (because 4·4 = 16)
The square root of 25 is 5 (because 5·5 = 25)
The square root of 100 is 10 (because 10·10 =100)

SCIENTIFIC NOTATION

For very large numbers, it is sometimes simpler to use "scientific notation" (so called, because scientifics often deal with very large numbers).

The format for writing a number in scientific notation is fairly simple:

First digit of the number followed by the decimal point and then all the rest of the digits of the number, times 10 to an appropiate power. The conversion is fairly simple.

Example: Write 12400 in scientific notation.

This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)·(10000)= 12400 and 10000= 104. Then, in scientific notation, 12400 is written as 1.24·104

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