 Some basics about exponents can be reviewed in
Chapter 1, section 4.
 Rules of Exponents

Rule

Example

x^{0} = 1

6^{0} = 1

x^{r}x^{s} = x^{r + s}

2^{3}2^{4} = 2^{3 + 4} = 2^{7}

x^{r} x^{s}

= x^{r  s}, x 0


6^{4} 6^{3}

= 6^{4  3} = 6^{1} = 6


(x^{r})^{s} = x^{rs}

(2^{3})^{4} = 2^{3·4} = 2^{12}

(xy)^{r} = x^{r}y^{r}

(2·6)^{2} = 2^{2}6^{2} = 4·36 = 144

(

x y

)

^{r}

=

x^{r} y^{r}

, y 0


(

6 2

)

^{2}

=

6^{2} 2^{2}

=

36 4

= 9


x^{r} =

1 x^{r}

, x 0



(

x y

)

^{r}

=

(

y x

)

^{r}

, x,y 0


(

2 6

)

^{2}

=

(

6 2

)

^{2}

= 9


 Scientific Notation
 Scientific Notation is used to express decimal numbers in a form such that
there is a number with one nonzero digit to the left of the decimal point
multiplied by an appropriate power of 10.

In Scientific Notation 
Not in Scientific Notation 
3.426 × 10^{6} 
3426000.0 
3.426 × 10^{6} 
0.000003426 
 Learn Rules by Example
 Example 1: Scientific notation to decimal
 3.1415 × 10^{7} = 3.1415 × 10000000 = 31415000.0
 The exponent (7) is positive.
 The decimal point is moved to the right 7 places.
 Exponent 1 means number 10.
 Example 2: Scientific notation to decimal
 3.1415 × 10^{7} = 3.1415 ÷ 10000000 = 0.00000031415
 The exponent (7) is negative.
 The decimal point is moved to the left 7 places.
 Exponent 1 means number is in the interval (0, 1).
 Example 3: Decimal to scientific notation
 123456789.0 = 1.23456789 × 100000000 = 1.23456789 × 10^{8}
 The decimal point had to be moved to the left 8 places until there was only one nonzero digit to the left of the decimal point.
 The exponent (8) is positive.
 Number 10 means exponent is 1.
 Example 4: Decimal to scientific notation
 0.0000123456 = 1.23456789 ÷ 100000 = 1.23456789 × 10^{5}
 The decimal point had to be moved to the right 5 places until there was only one nonzero digit to the left of the decimal point.
 The exponent (5) is negative.
 Number in the interval (0, 1) means exponent 1.
 Example 5: Decimal numbers in the interval [1, 10)
 2.34 = 2.34 × 10^{0} = 2.34
 2.34 is already in scientific notation
 Rules in Words

 Scientific Notation to Decimal
 Observe the exponent on the base 10.
(a) 
If the exponent is positive, move the decimal point to the right the same number of places as the exponent.
 You may have to add zeros to the number.
 This will result in a number 10.

(b) 
If the exponent is 0, do not move the decimal point.
 Drop the factor 10^{0}.
 This will result in a number in the interval [1, 10).

(c) 
If the exponent is negative, move the decimal point to the left
the same number of places as the absolute value of the exponent.
 You may have to add zeros to the number.
 This will result in a number in the interval (0, 1).

 Decimal to Scientific Notation
 Move the decimal point in the number to the right of the first nonzero digit.
 This results in a number in the interval [1, 10).
 Count the number of places you moved the decimal point in step 1.
 If the original decimal number is 10 or greater, the count is positive.
 If the original decimal number is in the interval (0, 1),
the count is negative.
 Multiply the number obtained in step 1 by 10 raised to the count (i.e., power)
found in step 2.
 Using Scientific Notation

 Idea
 Write given numbers in scientific notation
 Use rules of exponents on the powers of 10.
 Multiplication

(2000000)(40000) 
= (2 × 10^{6})(4 × 10^{4})
= (2)(4) × (10^{6})(10^{4})
= 8 × 10^{6 + 4}
= 8 × 10^{10}
= 80000000000

 Division

40000 0.002 
= 
4 × 10^{4} 2 × 10^{3} 

= 


= 2 × 10^{4  (3)}
= 2 × 10^{7}
= 20000000

