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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 

x^{2 < 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. 

5. To discover the average between two evenly spaced numbers, add 

the first and the last terms and divide them by 2.