Square of a Number
Squaring a number means to multiply that number by itself.
The notation for squaring a number (x) is as follows: x2
When squaring an integer, the result obtained is called a perfect
square.
When preparing for the test, make sure that you are fully capable of
understanding and reproducing the following table, as well as
recognizing the numbers that are perfect squares and perfect cubes.
12 = 1, 13 = 1
22 =4, 23 = 8
32 =9, 33 = 27
42 =16, 43 = 64
52 =25, 53 = 125
62 = 36, 63 = 216
72 = 49, 73 = 343
You will not need to know any higher.
Squared numbers and special properties
x2 > 0 always, except for x = 0
x2 > x for x > 1
x2 < x for 0 < x < 1
*important
x2 = x for x = 1 or 0
The square root of x2 equals the absolute value of x.
If x2= y2, then either x = y, or y = -x, or x = -y.
The following is an example:
Of the following numbers, which is a both a perfect square and a
perfect cube?
A. 4
B. 8
C. 9
D. 16
E. 64
Note: the answer is (E)
Exponents
The mathematical notations for numbers which are the result of a
number that is multiplied by itself a number of times is called
exponents.
Examples:
x3 = x × x × x
x5 = x × x × x × x × x
The expression of x n is also called the n th power of x. The x is the
base, while the n is the exponent. Math questions will usually only
utilize integral exponents. x2 is read as x-squared, and x3 is read as x-
cubed. All others are read as a power of x. x4 is read as the 4th power
of x.
When it comes to the power of 10, there is a simple, quick rule that
simplifies the powers of 10, by writing it as 1, followed by the number
of zeros as specified by the power.
Examples: 10 5 = 1 followed by 5 zeros. 100000 = 100,000.
An example you may find is:
Represent 32,456 to the power of 10.
The solution would be as follows:
32,456 = 3 × 10 4 + 2 × 10 3 + 4 × 10 2 + 5 × 10 1 + 6 × 100
Consider the following example:
Solve for x: (x - 3)2 = 49.
You could use algebra and take the square root of both sides or since
49 is a perfect square you could guess integers for x. Just remember x
-3 must be positive or negative.
If you try guessing, the integers 10 and -4 work. To get an algebra
solution, do the following:
(x - 3)2 = 49
x - 3 = 7 or x - 3 = -7
x = 10 or x = -4
It is your goal to get problems correct quickly. Sometimes guessing
(Guessing in this case means substituting in numbers to see which
satisfy the equation.) is faster than solving an equation, if you train
yourself to use the technique. Of course, if you cannot "see" the
answers fast enough, use other approaches to answer the problem.
Roots
The test will require you to manipulate both square roots and cube
roots. Some of the questions will measure whether or not you
understand these expressions.
You should remember that none of the following should ever occur:
1. No perfect square can be left underneath a radical (square root)
sign.
2. No radical can be within the denominator.
3. No fractions may occur within the radical sign.
Averages
There are three basic components that comprise an average problem:
1. Total
2. Average (also known as a mean)
3. # of numbers
The average is the total of elements that are within the set.
To discover the average, simply divide the total by the # of numbers.
For example:
Jenna's last four test scores were 35, 56, 75, and 28. What is the
average of Jenna's test scores?
A. 43
B. 48.5
C. 52.5
D. 54
E. 47
Note: the answer is (B).
35 + 56 + 75 + 28 = 194
194 / 4 = 48.5
Five things to remember when solving averages:
1. If a number that is the same as the average is added, the new
average will not change.
2. If a number is added and it is less than the average, the average
will decrease.
3. If a number is added and it is greater than the average, the
average will increase.
4. If a pair of numbers are added, and they are "balanced" on both
sides of the average, the
arithmetic mean is the middle value.
