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.
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