Division and Exponents

Division

Multiplication and division are inverse operations. We arrive at division facts by knowing multiplication facts. There are three terms to describe division: dividend, divisor, and quotient.

    

                                         12   ¸   4   =   3

                                     dividend        divisor     quotient

Division of Whole Numbers

For any whole number r and s, with  s ¹ 0, the quotient of r divided by s, written    r ¸ s, is the whole number k, if it exists, such that r = s x k.

Example:

               18 ¸ 3 = 6 because 3 x 6 = 18

Please visit the website Itcoline.net if you want any extra information about this subject.

True and False Statements:

         A.  5 ¸ 0 = 0  false since  0 x 0 ¹ 5

         B.  0 ¸ 5 = 0   true since   5 x 0 = 0

         C.  0 ¸ 0 = 0   false since  0 x ? = 0   0 ¸  0  =  any number   many answers (not a unique solution).



Concepts of Division

There are two concepts of division: The sharing (partitive) concept and the measurement (subtractive) concept.

Sharing/partitive

Illustrating the sharing/partitive concept of division. Suppose you have 30 colored chips, which you want to divide among 6 friends. How many colored chips would each friend receive?
          30 ¸  6 = 5

                       1    2    3    4    5    6

                        I     I     I     I     I     I                      
                        I     I     I     I     I     I          

                        I     I     I     I     I     I         five each      
                        I     I     I     I     I     I

                        I     I     I     I     I     I

Measurement/subtractive

Illustrating the measurement/subtractive concept of division. Suppose you have 30 colored chips and want to give 6 colored chips to as many friends as possible. How many friends would receive colored chips?

            30 ¸ 6 = 5

                              1   I I I I I I

                              2   I I I I I I

                              3   I I I I I I          five friends

                              4   I I I I I I

                              5   I I I I I I

                                 
 Division Theorem

For any whole numbers a and b, with divisor b ¹  0, there are whole numbers q (quotient) and r (remainder) such that
                                                divisor

                                         a  =   bq   +   r remainder
                                     dividend      quotient

and 0 £ r < b.

Example:

               5 x 2 + 2 = 12    «    12 ¸ 5 = 2 r 2

The remainder r is always less than the divisor b. If r = 0, then the quotient a ¸ b is the whole number q.



Mental Calculations and Estimation

Equal Quotients

Multiply or divide both the dividend and divisor by the same number.

Examples:

           1.    400 ¸ 16 = (400 ¸ 4) (16 ¸ 4) = 100 ¸ 4 = 25

           2.    77 ¸ 21 = (77 ¸ 7) (21 ¸ 7) = 11 ¸ 3 = 3 r 2


Compatible Numbers

Using compatible numbers is useful for mental estimating quotients (round to easy numbers).

Example:

              92 ¸ 9 » 90 ¸ 9 = 10

Rounding
Rounding is one method of estimating a quotient.

             145 ¸ 23 » 150 ¸ 25 = 6

Front-End Estimation
This technique can be used to obtain an estimated quotient of two numbers by using the leading digit of each number. The following example use both the compatible numbers and front-end estimation.

             783 ¸ 244  » 700 ¸ 200 = 7  ¸ 2 = 3 ½

Exponents

Exponentiation
The operation of raising numbers to a power is called exponentiation. A number in the form bⁿ is said to be in exponential form;  bⁿ  is called the nth power of  b. For any number b and any whole number n, with b and n not both zero, 

                                          bⁿ  = b x b x b x b x b x . . . x b     

                                                         b occurs n times

 Where  b  is  called  the  base  and  n  is  called  the  exponent.  in case  n = 0  or  n = 1,  b⁰ = 1 and  b¹ = b.

                                                                                                               

Laws of Exponents

For any number a and all whole numbers m and n, except for the case where the base and exponents are both zero,

              aⁿ x a^m  = a^{n + m}

                 aⁿ ¸ a^m  = a^{n - m}         for a ¹ 0

For more information about law of exponents go to the Oak Road Systems website.

Evaluating Exponential Expressions

Multiplication and division can be performed easy with numbers that are written as powers of the same base. To multiply, we add the exponents; to divide, we subtract the exponents.

Examples:

            1.    3²² x 3⁸ = 3³⁰

            2.    7³ x 7⁸ = 7¹¹

               3.    4⁶ ¸ 4² = 4⁴

               4.    10³²¸  10¹⁵ = 10¹⁷

For repeated multiplication we use exponents:

                                           3⁵ = 3 x 3 x 3 x 3 x 3

                                           6⁴ = 6 x 6 x 6 x 6

Order of Operations

Parentheses (or grouping symbols) first;  then numbers raised to a power (exponents) are evaluated; next products and quotients (multiplication/division) are computed from left to right; finally sums and differences are calculated from left to right.

Most of the information on this blog was taken from the text book: Mathematics for Elementary Teachers and from other sources.