The order in which mathematical operations are performed makes a difference in the outcome of the operations. To ensure that all mathematicians, and students, get the same answers to mathematical equations, a particular order of operations has been agreed upon. Operations inside parenthesis are always performed from the inside out. This means, operations inside parenthesis are performed before any other operations. Exponentiation (raising numbers to powers) is performed next. Multipication and division are done next in order. Addition and subtraction are always done last.
A mnemonic is an abbreviation of a series of words or phrases that is itself a word or phrase that can be more easily remembered than the series of words or phrases. The mnemonic for the order of operations is a nonsense word, PEMDAS, representing the first letter of the operations in the correct order. A phrase that helps keeps the operations in order, and is a better mnemonic, is Please Excuse My Dear Aunt Sally. Apparently Dear Aunt Sally needs excusing for some reason, possibly having to do with her mental state, but the first letters of the words in the phrase are the first letters of the operations in the correct order.
Now that we have a standard for the order of operations that we can all follow, let's look at some equations and see what properties these equations lead us to.
The Commutative Property
1. What is the sum of 3 and 4? What is the sum of 4 and 3? Is there a difference? Can you state a principle that would describe this property of real numbers? This property is called the commutative property of addition. Does this property hold if you take a difference of two numbers?
2. What is the product of 3 and 4? What is the product of 4 and 3? is there a difference? Can you state a principle that would describe this property of real numbers? This property is called the commutative property of multiplication. Does this property hold for the division of two numbers?
The Associative Property
3. What is the sum of 2, 3, and 4? We can only add two numbers at a time so we must pick two numbers of the three to add first before adding the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you state a principle that would describe this property of real numbers? This property is called the associative property of addition. Does this property work if one or more of the operations is a difference?
4. What is the product of 2, 3, and 4? We can only multiply two numbers at a time so we must pick two numbers of the three to multiply first before multiplying the third. This requires us to group two of the numbers together. Does it matter which two numbers we group together first? Can you state a principle that would describe this property of real numbers? This property is called the associative property of multiplication. Does this property work if one or more of the operations is a division?
The Distributive Property
5. Calculate the result of the following equation: 2×(3+4). Calculate the result of the following equation: 6+8. How are these two equations related? What happened to the multiplication and the addition between the two equations? This property of real numbers is called the distributive property and is the agreed upon method for using multiplication and addition together. Can you undo the number 27 so that you have the product of a number and a sum like the first equation? This is the process of undoing a distribution and is called factoring. We will make extensive use of this property to combine a product over a sum and to undo a product over a sum.
6. Can you use the distributive property to properly distribute (2+3)×(4+5)? How would this look if you wrote it in distributed form like the second equation above? Hint: Multiplication of large numbers uses a method that stacks large numbers and keeps powers of ten in the same columns. A large number is actually a sum of powers of ten, a multiple of a hundred plus a multiple of ten plus a multiple of one for example. Stacking the numbers and multiplying in columns helps properly perform the distributive property. Try doing something similar with (2+3)×(4+5) and see if you get the right answer.
The Identities
7. Can you think of a number that when added to any other number doesn't change the value of that second number? This number is called the additive identity because adding this number doesn't change a number's identity.
8. Can you think of a number that when multiplied to any other number doesn't change the value of the second number? This number is called the multiplicative identity because multiplying this number doesn't change a number's identity. We will use this property to change the appearance of rational numbers without changing the rational numbers' values. Any rational number whose numerator and denominator are the same are equivalent to the multiplicative identity. Multiplying a rational number by a rational representation of the multiplicative identity changes the appearance of a rational number but doesn't change its value.
The Inverses
9. Can you think of an integer that when added to 3 will give you the additive identity? This integer is 3's additive inverse. Can you think of an integer that when added to −4 will give you the additive identity? This number is −4's additive inverse. Can you state a principle that will allow you to find any real number's additive inverse? Adding a number and its additive inverse to a number or equation adds the additive identity to the number or equation. This is called adding a well chosen zero and we will find many situations when this will become advantageous.
10. Can you think of a rational number that when multiplied to 3/4 will give you the multiplicative identity? This rational number is 3/4's multiplicative inverse. Can you think of a rational number that when multiplied to 5/4 gives you the multiplicative identity? This rational number is 5/4's multiplicative inverse. Can you state a principle that would allow you to find the multiplicative inverse of any real number? The multiplicative inverse is called the reciprocal.
The Laws of Exponents
Exponentiation, or raising a number to a power, is multiple multiplications. In the number 23, 2 is the base and 3 is the exponent. The exponent tells you how many times to multiply the base to itself. A number raised to the 2nd power is said to be squared. A number raised to the 3rd power is said to be cubed. Any number without an exponent is understood to have an exponent of 1. Any number raised to the 0 power equals the multiplicative identity.
Let's look at some consequences of this description of exponentiation.
11. Using the multiple multiplications of exponentiation, can you find a way to rewrite the product of 22 and 23 as a single exponent? Can you state a principle that would allow you to describe this product of two exponentiated numbers? Does it work for the product of 23 and 32? Can you make your principle more specific?
12. Using the multiple multiplications of exponentiation, can you find a way to rewrite 22 cubed using a single exponent? Can you state a principle that would allow you to describe this exponentiation of an exponent? Why do you get the same answer for 23 squared?
13. Using the multiple multiplications of exponentiation, can you find a way to rewrite (2×3)3 as a product of two numbers each raised to a power? Can you state a principle that would allow you to express a product raised to a power as a product of two numbers each raised to a power?
14. Use the multiple multiplications of exponentiation and the rational representation of the multiplicative identity to simplify 2 cubed over 2 squared. Can you state a principle that would allow you to rewrite this rational expression using a single exponent? Does your principle work for the rational number 2 sqaured over 2 cubed? Can you state a principle for negative exponents?
15. Using the multiple multiplications of exponentiation, can you rewrite 2 cubed over 3 cubed as a rational number raised to a power? Can you state a principle that would allow you to rewrite any rational number in which the numerator and denominator are raised to a power as a rational number raised to a power? Can you apply a previous principle and this principle to see what happens to 2 cubed over 3 to the sixth?
3 is the 3rd root of 27 if and only if 33=27. In 4√23, 4 is the index, 2 is the base and 3 is still an exponent. The 2nd root is called the square root. Any radical without an index is under stood to be the square root. The 3rd root is called the cube root.
To reduce a radical, factor the base into its prime number factors, then, using the index, remove groups of factors by the index until you can't remove any more. Each time you remove a group, you count the base once on the outside of the radical. When you can remove no more groups, the remaining factors are multiplied back together again and the bases on the outside are multiplied together. Any radical with no base left inside it is a whole integer number. Any radical with a base left inside it is an irrational number. Most radicals are irrational numbers.
Using this description of radicals as a basis, let's look at some consequences.
16. Simplify 3√23. Simplify 3√26. What would you do with 3√28? Can you state a principle for the case where the index and the exponent on the base are the same?
17. Simplify 3√23×3√33. Using the definition of a root, rewrite these two numbers a product under a single radical. Can you state a principle that would describe this radical. Do the bases have to be the same? Do the indexes have to be the same?
A radical can be written as a rational exponent. The exponent of the number under the radical is the numerator, and the index of the radical is the denominator. Express 5√26 as a rational exponent. Using rational exponents for 17, do the indexes still have to be the same?
