Mathematic vocabulary

1. Multiplying and Dividing Powers

To multiply powers with the same base, add the exponents and keep the same base:

3                 4          3+4               7

b     X     b =    b         =   b

To divide powers with the same base, subtract the exponents and keep the same base:

12          8             12-8               4

b   /     b   =     b        =     b

2. Raising Powers to Powers

To raise a power to a power, multiply the exponents:

3    5         3×5              15

(x   )  =   x      =     x

3. Negative Powers

A number raised to a negative exponent is simply the reciprocal of that number raised to the corresponding positive exponent.

-3

2   = 1 =  1

3       8

2

4. Simplifying Square Roots ________

V

To simplify a square root, factor out the perfect squares under the radical, unsquare them and put the result in front:

__       ____     __         __           ___

V12 = V 4X3 = V 4   X  V 3    = 2 V 3

5. Adding and Subtracting Roots

You can add or subtract radical expressions when the part under the radicals is the same:

__        __       ___

2 V7 + 3 V7 = 5 V7

Don’t try to add or subtract when the radicals are different.  You cannot simplify expressions like:          ___             __

2  V 3     +  3  V 5

6. Multiplying and Dividing Roots

The product of square roots is equal to the square root of the product:

__          __       ______       ___

V 2    x  V 3  =  V 2 X  3  =  V 6

The quotient of square roots is equal to the square root of the quotient:

__            ___         ____        ___

V 8    /     V  2     =  V 8/4   =  V 2

1. Integers

Integers are whole numbers.. .-4,-3,-2,-1,0, 1,2,3,4,5…….

Positive integers are the numbers 1,2,3,4,5….

Zero is neither positive nor negative.

Negative integers are the numbers -1,-2,-3,-4,-5,-6,-7

Consecutive integers are writeen as x, x+1, x+2,….

Consecutive even or odd integers are written as x, x+2, x+4, x+6,…..

2. Nonintegers

Nonintegers are numbers which have a fractional part.

Examples of nonintegers are t, 3.75, -1/2, 5/6 and pi.

3. Adding/Subtracting Signed Numbers

To add a positive and a negative, first ignore the signs and find the positive difference between the number parts. Then attach the sign of the original number with the larger number part.

For example, to add 41 and -28, first we ignore the minus sign and find the positive difference between 41 and 28,which is 13. Then we attach the sign of the number with the larger number part.  In this case it’s the plus sign from the 41.   So, 41 + (-28) = 13.

Make subtractions simpler by turning them into addition. For example, think of

-18 -(-26) as -18 + (+26).

To add or subtract a string of positives and negatives, first turn everything into addition. Then

combine the positives and negatives so that the string is reduced to the sum of a single positive

number and a single negative number.

4. Multiplying/Dividing Signed Numbers

To multiply and/or divide positives and negatives, treat the numbes as usual and attach a minus sign if there were originally an odd number of negatives.

For example, to multiply -2, -4, and -6, first multiply the number parts:

2 X 4 X 6 = 30.   Then go back and note that there were three negatives (an odd number), so the

product is negative: (-2) X (-4) X (-6) = -48.

5. Order of Operations

Perform multiple operations in the following order:

a)  Parentheses

b)  Exponents

c)  Multiplication and Division (left to right)

d)  Addition and Subtraction (left to right)

In the expression 9 -3 X (6 -3) + 6/3 , begin with the parentheses: (6 -3) = 3. Then do the exponent: (3)(3) = 9.   Now the expression is: 9 -3 X 9 + 6/3.   Next do the multiplication and division to get: 9 – 21 + 2, which equals -10.

6. Counting Consecutive Integers

To count consecutive integers, subtract the smallest from the largest and add 1. To count the

integers from 18 through 56, subtract: 56 -18 = 38.   Then add 1: 38 + 1 = 39.

7. Absolute Value

The absolute value of any number is its distance from zero on the number line. The absolute value of a positive number is simply that number. To find the absolute value of a negative number, just drop the negative sign. Absolute value is represented by putting two vertical lines around the number. So the absolute value of 8 = /8/ = 8. The absolute value of -43 = /-43/ = 43. The absolute value of any nonzero number is always positive. The absolute value of 0 is 0.

1. Average or Arithmetic Mean

To find the average of a set of numbers, add them up and divide by the number of numbers.

Sum of the terms

Average                        =         Number of terms

To find the average of the five numbers 12, 15, 23, 40, and 40, first add them:

12 + 15 + 23 + 40 + 40 = 130. Then divide the sum by 5: 130 / 5 = 26.

2. Using the Average to Find the Sum

Sum = (Average) X (Number of terms)

If the average of ten numbers is 60, then they add up to 10 X 60, or 600.

3. Finding a Missing Number

To find a missing number when you’re given the average, use the sum.   If the average of four numbers is 7, then the sum of those four numbers is 4  X  7, or 28.   Suppose that three of the numbers are 3, 5, and 8. These three numbers add up to 16 of that 28, which leaves 12 for the fourth number.

4. Median

The median of a set of numbers is the value that falls in the middle of the set. If you have five test scores, and they are 88, 86, 57, 94, and 73, you must first list the scores in increasing or  decreasing order: 57,73, 86, 88, 94.

The median is the middle number, or 86. If there is an even number of values in a set (six test scores, for instance), simply take the average of the two middle numbers.

5. Mode

The mode of a set of numbers is the value that appears most often.   If your test scores were 88, 57, 68, 85,99, 93, 93, 84, and 81, the mode of the scores would be 93 because it appears more often than any other score.  If there is a tie for the most common value in a set, the set has more than one mode.

6. Standard Deviation

Standard Deviation is a complex statistical measure, but for the test you mainly need to know that the it is the measure of how spread out a group of numbers are.  For example, the numbers {0, 10, 20} have a Standard Deviation of about 8.17 while the numbers {9, 10, 11} have a Standard Deviation of about 0.82.   Both have an average of 10, but because the first group was more “spread out” it had a higher Standard Deviation.