Properties of Basic Mathematical Operations

Some mathematical operations have properties that can make them easier to work with and can actually save you time.

Some properties (axioms) of addition

You should know the definition of each of the following properties of addition and how each can be used.

• Closure is when all answers fall into the original set. If you add two even numbers, the answer is still an even number (2 + 4 = 6); therefore, the set of even numbers is closed under addition (has closure). If you add two odd numbers, the answer is not an odd number (3 + 5 = 8); therefore, the set of odd numbers is not closed under addition (no closure).
• Commutative means that the order does not make any difference in the result.
  
  
  Note: Commutative does not hold for subtraction.
  
  
• Associative means that the grouping does not make any difference in the result.
  
  
  The grouping has changed (parentheses moved), but the sides are still equal.
  Note: Associative does not hold for subtraction.
  
  
• The identity element for addition is 0. Any number added to 0 gives the original number.
  
  
• The additive inverse is the opposite (negative) of the number. Any number plus its additive inverse equals 0 (the identity).
  
  

Some properties (axioms) of multiplication

You should know the definition of each of the following properties of multiplication and how each can be used.

• Closure is when all answers fall into the original set. If you multiply two even numbers, the answer is still an even number (2 × 4 = 8); therefore, the set of even numbers is closed under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number (3 × 5 = 15); therefore, the set of odd numbers is closed under multiplication (has closure).
• Commutative means the order does not make any difference.
  
  
  Note: Commutative does not hold for division.
  
  
• Associative means that the grouping does not make any difference.
  
  
  The grouping has changed (parentheses moved) but the sides are still equal.
  Note: Associative does not hold for division.
  
  
• The identity element for multiplication is 1. Any number multiplied by 1 gives the original number.
  
  
• The multiplicative inverse is the reciprocal of the number. Any nonzero number multiplied by its reciprocal equals 1.
  ; therefore, 2 and are multiplicative inverses.
  ; therefore, a and are multiplicative inverses (provided a 0).

A property of two operations

The distributive property is the process of passing the number value outside of the parentheses, using multiplication, to the numbers being added or subtracted inside the parentheses. In order to apply the distributive property, it must be multiplication outside the parentheses and either addition or subtraction inside the parentheses.
Note: You cannot use the distributive property with only one operation.

Multiplying and Dividing Using Zero

Zero times any number equals zero.
Likewise, zero divided by any nonzero number is zero.
Important note: Dividing by zero is “undefined” and is not permitted.
and are not permitted.
has no answer and does not have a unique answer.
In neither case is the answer zero.

Powers and Exponents

An exponent is a positive or negative number placed above and to the right of a quantity. It expresses the power to which the quantity is to be raised or lowered. In 4³, 3 is the exponent and 4 is called the base. It shows that 4 is to be used as a factor three times. 4 × 4 × 4 (multiplied by itself twice). 4³ is read as four to the third power (or four cubed).
Remember that x¹ = x and x⁰ = 1 when x is any number (other than 0).
If the exponent is negative, such as 3^–2, then the base can be dropped under the number 1 in a fraction and the exponent made positive. An alternative method is to take the reciprocal of the base and change the exponent to a positive value.

Squares and cubes

Two specific types of powers should be noted, squares and cubes. To square a number, just multiply it by itself (the exponent would be 2). For example, 6 squared (written 6²) is 6 × 6, or 36. 36 is called a perfect square (the square of a whole number). Following is a list of the first twelve perfect squares:
To cube a number, just multiply it by itself twice (the exponent would be 3). For example, 5 cubed (written 5³) is 5 × 5 × 5, or 125. 125 is called a perfect cube (the cube of a whole number). Following is a list of the first twelve perfect cubes.

Operations with powers and exponents

To multiply two numbers with exponents, if the base numbers are the same, simply keep the base number and add the exponents.
To divide two numbers with exponents, if the base numbers are the same, simply keep the base number and subtract the second exponent from the first, or the exponent of the denominator from the exponent of the numerator.
(Some shortcuts are possible.)
To add or subtract numbers with exponents, whether the base numbers are the same or different, you must simplify each number with an exponent first and then perform the indicated operation.
If a number with an exponent is raised to another power (4²)³, simply keep the original base number and multiply the exponents.

Example 6

Multiply and leave the answers with exponents.

1. 
   
2. 

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By Muhammad Atif Saeed on 00:04. Filed under Mathandstat . Follow any responses to the RSS 2.0. Leave a response