Wonderlic Mathematics Review: Operations

There are four basic mathematical operations:

Addition increases the value of one quantity by the value of another quantity.

Example: .  The result is called the sum.  With addition, the order does not matter. .

Subtraction is the opposite operation to addition; it decreases the value of one quantity by the value of another quantity.

Example: . 
The result is called the difference. 

Note that with subtraction, the order does matter.  .

Multiplication can be thought of as repeated addition.  One number tells how many times to add the other number to itself. 

Example: .  With multiplication, the order does not matter.   or .

Division is the opposite operation to multiplication; one number tells us how many parts to divide the other number into. 

Example: ; if 20 is split into 4 equal parts, each part is 5.  With division, the order of the numbers does matter. .

Subtraction, Multiplication, and Division

An exponent is a superscript number placed next to another number at the top right.  It indicates how many times the base number is to be multiplied by itself.  Exponents provide a shorthand way to write what would be a longer mathematical expression. 
Example: ; .  A number with an exponent of 2 is said to be “squared,” while a number with an exponent of 3 is said to be “cubed.”  The value of a number raised to an exponent is called its power.  So, 84 is read as “8 to the 4th power,” or “8 raised to the power of 4.”  A negative exponent is the same as the reciprocal of a positive exponent. Example: .

Parentheses are used to designate which operations should be done first when there are multiple operations.  Example: 4 – (2 + 1) = 1; the parentheses tell us that we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an answer of 3).

Order of Operations is a set of rules that dictates the order in which we must perform each operation in an expression so that we will evaluate it accurately. 

If we have an expression that includes multiple different operations, Order of Operations tells us which operations to do first. 

The most common mnemonic for Order of Operations is PEMDAS, or "Please Excuse My Dear Aunt Sally." 

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.  It is important to understand that multiplication and division have equal precedence, as do addition and subtraction, so those pairs of operations are simply worked from left to right in order.
Example: Evaluate the expression  using the correct order of operations.
P:  Perform the operations inside the parentheses, .
E:  Simplify the exponents, .

The equation now looks like this: .

MD:  Perform multiplication and division from left to right,
; then .

The equation now looks like this: .
AS:  Perform addition and subtraction from left to right, ; then .

The laws of exponents are as follows:

1. Any number to the power of 1 is equal to itself: .

2. The number 1 raised to any power is equal to 1: .

3. Any number raised to the power of 0 is equal to 1: .

4. Add exponents to multiply powers of the same base number:.

5. Subtract exponents to divide powers of the same number; that is .

6. Multiply exponents to raise a power to a power: .

7. If multiplied or divided numbers inside parentheses are collectively raised to a power, this is the same as each individual term being raised to that power: ; .

Note: Exponents do not have to be integers.  Fractional or decimal exponents follow all the rules above as well.  Example: .

A root, such as a square root, is another way of writing a fractional exponent.  Instead of using a superscript, roots use the radical symbol () to indicate the operation. 

A radical will have a number underneath the bar, and may sometimes have a number in the upper left: , read as “the nth root of a.” 

The relationship between radical notation and exponent notation can be described by this equation: . 

The two special cases of n = 2 and n = 3 are called square roots and cube roots. 

If there is no number to the upper left, it is understood to be a square root (). 

Nearly all of the roots you encounter will be square roots. 

A square root is the same as a number raised to the one-half power. 

When we say that a is the square root of b (), we mean that a multiplied by itself equals b: ().

A perfect square is a number that has an integer for its square root. 

There are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1 through 10).

Scientific notation is a way of writing large numbers in a shorter form. 

The form  is used in scientific notation, where a is greater than or equal to 1, but less than 10, and n is the number of places the decimal must move to get from the original number to a. 

Example: The number 230,400,000 is cumbersome to write. 

To write the value in scientific notation, place a decimal point between the first and second numbers, and include all digits through the last non-zero digit (). 

To find the appropriate power of 10, count the number of places the decimal point had to move (). 

The number is positive if the decimal moved to the left, and negative if it moved to the right.  We can then write 230,400,000 as . 

If we look instead at the number 0.00002304, we have the same value for a, but this time the decimal moved 5 places to the right (). 

Thus, 0.00002304 can be written as . 

Using this notation makes it simple to compare very large or very small numbers. 

By comparing exponents, it is easy to see that  is smaller than , because 4 is less than