Arithmetic for Praxis I: Pre-Professional Skills Test Study Guide (page 6)

This section covers the basics of mathematical operations and their sequence. It also reviews variables, integers, fractions, decimals, and square roots.

Numbers and Symbols

Numbers and the Number Line

• Counting numbers (or natural numbers): 1, 2, 3,…
• Whole numbers include the counting numbers and zero: 0, 1, 2, 3, 4, 5, 6,…
• Integers include the whole numbers and their opposites. Remember, the opposite of zero is zero:…–3, –2, –1, 0, 1, 2, 3,…
• Rational numbers are all numbers that can be written as fractions, where the numerator and denominator are both integers, but the denominator is not zero. For example, is a rational number, as is . The decimal form of these numbers is either a terminating (ending) decimal, such as the decimal form of which is 0.75; or a repeating decimal, such as the decimal form of , which is 0.3333333…
• Irrational numbers are numbers that cannot be expressed as terminating or repeating decimals (i.e., nonrepeating, nonterminating decimals such as π, ).

The number line is a graphical representation of the order of numbers. As you move to the right, the value increases. As you move to the left, the value decreases.

If we need a number line to reflect certain rational or irrational numbers, we can estimate where they should be.

Comparison Symbols

The following table will illustrate some comparison symbols:

Symbols of Addition

In addition, the numbers being added are called addends. The result is called a sum. The symbol for addition is called a plus sign. In the following example, 4 and 5 are addends and 9 is the sum:

4 + 5 = 9

Symbols of Subtraction

In subtraction, the number being subtracted is called the subtrahend. The number being subtracted from is called the minuend. The answer to a subtraction problem is called a difference. The symbol for subtraction is called a minus sign. In the following example, 15 is the minuend, 4 is the subtrahend, and 11 is the difference:

15 – 4 = 11

Symbols of Multiplication

When two or more numbers are being multiplied, they are called factors. The answer that results is called the product. In the following example, 5 and 6 are factors and 30 is their product:

5 × 6 = 30

There are several ways to represent multiplication in this mathematical statement.

• A dot between factors indicates multiplication:

5 · 6 = 30

• Parentheses around any one or more factors indicate multiplication:

(5)6 = 30, 5(6) = 30, and (5)(6) = 30.

• Multiplication is also indicated when a number is placed next to a variable: 5a = 30. In this equation, 5 is being multiplied by a.

Symbols of Division

In division, the number being divided by is called the divisor. The number being divided into is called the dividend. The answer to a division problem is called the quotient.

There are a few different ways to represent division with symbols. In each of the following equivalent expressions, 3 is the divisor and 8 is the dividend:

Prime and Composite Numbers

A positive integer that is greater than the number 1 is either prime or composite, but not both.

• A prime number is a number that has exactly two factors: 1 and itself.
  Examples
  2, 3, 5, 7, 11, 13, 17, 19, 23…
• A composite number is a number that has more than two factors.
  Examples
  4, 6, 8, 9, 10, 12, 14, 15, 16…
• The number 1 is neither prime nor composite since it has only one factor.

Operations

Addition

Addition is used when it is necessary to combine amounts. It is easiest to add when the addends are stacked in a column with the place values aligned. Work from right to left, starting with the ones column.

Example

Add 40 + 129 + 24.

1. Align the addends in the ones column. Because it is necessary to work from right to left, begin to add starting with the ones column. The ones column totals 13, and 13 equals 1 ten and 3 ones, so write the 3 in the ones column of the answer, and regroup or "carry" the 1 ten to the next column as a 1 over the tens column so it gets added with the other tens:
   
2. Add the tens column, including the regrouped 1.
   
3. Then add the hundreds column. Because there is only one value, write the 1 in the answer.
   

Subtraction

Subtraction is used to find the difference between amounts. It is easiest to subtract when the minuend and subtrahend are in a column with the place values aligned. Again, just as in addition, work from right to left. It may be necessary to regroup.

Example

If Becky has 52 clients and Claire has 36, how many more clients does Becky have?

1. Find the difference between their client numbers by subtracting. Start with the ones column. Because 2 is less than the number being subtracted (6), regroup or "borrow" a ten from the tens column. Add the regrouped amount to the ones column. Now subtract 12 – 6 in the ones column.
   
2. Regrouping 1 ten from the tens column left 4 tens. Subtract 4 – 3 and write the result in the tens column of the answer. Becky has 16 more clients than Claire. Check by addition: 16 + 36 = 52.
   

Multiplication

In multiplication, the same amount is combined multiple times. For example, instead of adding 30 three times, 30 + 30 + 30, it is easier to simply multiply 30 by 3. If a problem asks for the product of two or more numbers, the numbers should be multiplied to arrive at the answer.

Example

A school auditorium contains 54 rows, each containing 34 seats. How many seats are there in total?

1. In order to solve this problem, you could add 34 to itself 54 times, but we can solve this problem more easily with multiplication. Line up the place values vertically, writing the problem in columns. Multiply the number in the ones place of the top factor (4) by the number in the ones place of the bottom factor (4): 4 × 4 = 16. Because 16 = 1 ten and 6 ones, write the 6 in the ones place in the first partial product. Regroup or carry the ten by writing a 1 above the tens place of the top factor.
   
2. Multiply the number in the tens place in the top factor (3) by the number in the ones place of the bottom factor (4): 4 × 3 = 12. Then add the regrouped amount: 12 + 1 = 13. Write the 3 in the tens column and the 1 in the hundreds column of the partial product.
   
3. The last calculations to be done require multiplying by the tens place of the bottom factor. Multiply 5 (tens from bottom factor) by 4 (ones from top factor); 5 × 4 = 20, but because the 5 really represents a number of tens, the actual value of the answer is 200 (50 × 4 = 200). Therefore, write the two zeros under the ones and tens columns of the second partial product and regroup or carry the 2 hundreds by writing a 2 above the tens place of the top factor.
   
4. Multiply 5 (tens from bottom factor) by 3 (tens from top factor); 5 × 3 = 15, but because the 5 and the 3 each represent a number of tens, the actual value of the answer is 1,500 (50 × 30 = 1,500). Add the two additional hundreds carried over from the last multiplication: 15 + 2 = 17 (hundreds). Write the 17 in front of the zeros in the second partial product.
   
5. Add the partial products to find the total product:
   

Note: It is easier to perform multiplication if you write the factor with the greater number of digits in the top row. In this example, both factors have an equal number of digits, so it does not matter which is written on top.

Division

In division, the same amount is subtracted multiple times. For example, instead of subtracting 5 from 25 as many times as possible, 25 – 5 – 5 – 5 – 5 – 5, it is easier to simply divide, asking how many 5s are in 25: 25 ÷ 5.

Example

At a road show, three artists sold their beads for a total of $54. If they share the money equally, how much money should each artist receive?

1. Divide the total amount ($54) by the number of ways the money is to be split (3). Work from left to right. How many times does 3 divide 5? Write the answer, 1, directly above the 5 in the dividend, because both the 5 and the 1 represent a number of tens. Now multiply: since 1(ten) × 3(ones) = 3(tens), write the 3 under the 5, and subtract; 5(tens) – 3(tens) = 2(tens).
   
2. Continue dividing. Bring down the 4 from the ones place in the dividend. How many times does 3 divide 24? Write the answer, 8, directly above the 4 in the dividend. Because 3 × 8 = 24, write 24 below the other 24 and subtract 24 – 24 = 0.
   

Remainders

If you get a number other than zero after your last subtraction, this number is your remainder.

Example
9 divided by 4.
1 is the remainder.
The answer is 2 r1. This answer can also be written as , because there was one part left over out of the four parts needed to make a whole.

Working with Integers

Remember, an integer is a whole number or its opposite. Here are some rules for working with integers:

Adding

Adding numbers with the same sign results in a sum of the same sign:

(positive) + (positive) = positive and (negative) + (negative) = negative

When adding numbers of different signs, follow this two-step process:

1. Subtract the positive values of the numbers. Positive values are the values of the numbers without any signs.
2. Keep the sign of the number with the larger positive value.

Example

–2 + 3 =

1. Subtract the positive values of the numbers: 3 – 2 = 1.
2. The number 3 is the larger of the two positive values. Its sign in the original example was positive, so the sign of the answer is positive. The answer is positive 1.

Example

8 + –11 =

1. Subtract the positive values of the numbers: 11 – 8 = 3.
2. The number 11 is the larger of the two positive values. Its sign in the original example was negative, so the sign of the answer is negative. The answer is negative 3.

Subtracting

When subtracting integers, change the subtraction sign to addition and change the sign of the number being subtracted to its opposite. Then follow the rules for addition.

Example
(+10) – (+12) = (+10) + (–12) = –2
(–5) – (–7) = (–5) + (+7) = +2

Multiplying and Dividing

A simple method for remembering the rules of multiplying and dividing is that if the signs are the same when multiplying or dividing two quantities, the answer will be positive. If the signs are different, the answer will be negative.

Example
(10)(–12) = –120
–5 × –7 = 35

Sequence of Mathematical Operations

There is an order in which a sequence of mathematical operations must be performed:

P: Parentheses/Grouping Symbols. Perform all operations within parentheses first. If there is more than one set of parentheses, begin to work with the innermost set and work toward the outside. If more than one operation is present within the parentheses, use the remaining rules of order to determine which operation to perform first.

E: Exponents. Evaluate exponents.

M/D: Multiply/Divide. Work from left to right in the expression.

A/S: Add/Subtract. Work from left to right in the expression.

This order is illustrated by the acronym PEMDAS, which can be remembered by using the first letter of each of the words in the phrase: Please Excuse My Dear Aunt Sally.

Example

Properties of Arithmetic

There are several properties of mathematics:

• Commutative Property: This property states that the result of an arithmetic operation is not affected by reversing the order of the numbers. Multiplication and addition are operations that satisfy the commutative property.
  Example
  5 × 2 = 2 × 5
  5a = a5
  b + 3 = 3 + b

However, neither subtraction nor division is commutative, because reversing the order of the numbers does not yield the same result.

Example
5 – 2 =≠ 2 – 5
6 ÷ 3 =≠ 3 ÷ 6
• Associative Property: If parentheses can be moved to group different numbers in an arithmetic problem without changing the result, then the operation is associative. Addition and multiplication are associative.
  Example
  2 + (3 + 4) = (2 + 3) + 4
  2(ab) = (2a)b
• Distributive Property: When a value is being multiplied by a sum or difference, multiply that value by each quantity within the parentheses. Then, take the sum or difference to yield an equivalent result.
  Example
  5(a + b) = 5a + 5b
  5(100 – 6) = (5 × 100) – (5 × 6)

This second example can be proved by performing the calculations:

5(94) = 5(100 – 6)
= 500 – 30
470 = 470

Additive and Multiplicative Identities and Inverses

• The additive identity is the value that, when added to a number, does not change the number. For all of the sets of numbers defined previously (counting numbers, integers, rational numbers, etc.), the additive identity is 0.
  Example
  5 + 0 = 5
  –3 + 0 = –3

Adding 0 does not change the values of 5 and –3, so 0 is the additive identity.

• The additive inverse of a number is the number that, when added to the number, gives you the additive identity.
  Example
  What is the additive inverse of –3?
  This means, "What number can I add to –3 to give me the additive identity (0)?"
  –3 + ___ = 0
  –3 + 3 = 0
  The answer is 3.
• The multiplicative identity is the value that, when multiplied by a number, does not change the number. For all of the sets of numbers defined previously (counting numbers, integers, rational numbers, etc.) the multiplicative identity is 1.
  Example
  5 × 1 = 5
  –3 × 1 = –3

Multiplying by 1 does not change the values of 5 and –3, so 1 is the multiplicative identity.

• The multiplicative inverse of a number is the number that, when multiplied by the number, gives you the multiplicative identity.

   

  Example
  What is the multiplicative inverse of 5?

   

  This means, "What number can I multiply 5 by to give me the multiplicative identity (1)?"

   

  5 × ___ = 1
  5 × = 1
  The answer is .

There is an easy way to find the multiplicative inverse. It is the reciprocal, which is obtained by reversing the numerator and denominator of a fraction. In the preceding example, the answer is the reciprocal of 5; 5 can be written as , so the reciprocal is .

Here are some numbers and their reciprocals:

Note: The additive inverse of a number is the opposite of the number; the multiplicative inverse is the reciprocal.

Factors and Multiples

Factors

Factors are numbers that can be divided into a larger number without a remainder.

Example
12 ÷ 3 = 4

The number 3 is, therefore, a factor of the number 12. Other factors of 12 are 1, 2, 4, 6, and 12. The common factors of two numbers are the factors that both numbers have in common.

Example
The factors of 24 = 1, 2, 3, 4, 6, 8, 12, and 24.
The factors of 18 = 1, 2, 3, 6, 9, and 18.

From the examples, you can see that the common factors of 24 and 18 are 1, 2, 3, and 6. From this list it can also be determined that the greatest common factor of 24 and 18 is 6. Determining the greatest common factor (GCF) is useful for simplifying fractions.

Example

The factors of 16 are 1, 2, 4, 8, and 16. The factors of 20 are 1, 2, 4, 5, and 20. The common factors of 16 and 20 are 1, 2, and 4. The greatest of these, the GCF, is 4. Therefore, to simplify the fraction, both numerator and denominator should be divided by 4.

Multiples

Multiples are numbers that can be obtained by multiplying a number x by a positive integer.

Example
5 × 7 = 35

The number 35 is, therefore, a multiple of the number 5 and of the number 7. Other multiples of 5 are 5, 10, 15, 20, and so on. Other multiples of 7 are 7, 14, 21, 28, and so on.

 

The common multiples of two numbers are the multiples that both numbers share.

 

Example
Some multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36…
Some multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48…

Some common multiples are 12, 24, and 36. From the above it can also be determined that the least common multiple of the numbers 4 and 6 is 12, since this number is the smallest number that appeared in both lists. The least common multiple, or LCM, is used when performing addition and subtraction of fractions to find the least common denominator.

Example (using denominators 4 and 6 and LCM of 12)

Decimals

The most important thing to remember about decimals is that the first place value to the right of the decimal point is the tenths place. The place values are as follows:

In expanded form, this number can also be expressed as:

1,268.3457 = (1 × 1,000) + (2 × 100) + (6 × 10) + (8 × 1) + (3 × .1) + (4 × .01) + (5 × .001) + (7 × .0001)

Adding and Subtracting Decimals

Adding and subtracting decimals is very similar to adding and subtracting whole numbers. The most important thing to remember is to line up the decimal points. Zeros may be filled in as placeholders when all numbers do not have the same number of decimal places.

Example
What is the sum of 0.45, 0.8, and 1.36?
Take away 0.35 from 1.06.

Multiplication of Decimals

Multiplication of decimals is exactly the same as multiplication of integers, except one must make note of the total number of decimal places in the factors.

Example
What is the product of 0.14 and 4.3?

 

First, multiply as usual (do not line up the decimal points):

 

Now, to figure out the answer, 4.3 has one decimal place and .14 has two decimal places. Add in order to determine the total number of decimal places the answer must have to the right of the decimal point. In this problem, there are a total of three (1 + 2) decimal places. When finished multiplying, start from the right side of the answer, and move to the left the number of decimal places previously calculated.

In this example, 602 turns into .602, because there have to be three decimal places in the answer. If there are not enough digits in the answer, add zeros in front of the answer until there are enough.

Example

 

Multiply 0.03 × 0.2.

 

There are three total decimal places in the problem; therefore, the answer must contain three decimal places. Starting to the right of 6, move left three places. The answer becomes 0.006.

Dividing Decimals

Dividing decimals is a little different from integers for the setup, and then the regular rules of division apply. It is easier to divide if the divisor does not have any decimals. In order to accomplish that, simply move the decimal place to the right as many places as necessary to make the divisor a whole number. If the decimal point is moved in the divisor, it must also be moved in the dividend in order to keep the answer the same as the original problem; 4 ÷ 2 has the same solution as its multiples 8 ÷ 4 and 28 ÷ 14. Moving a decimal point in a division problem is equivalent to multiplying a numerator and denominator of a fraction by the same quantity, which is the reason the answer will remain the same.

If there are not enough decimal places in the answer to accommodate the required move, simply add zeros until the desired placement is achieved. Add zeros after the decimal point to continue the division until the decimal terminates, or until a repeating pattern is recognized. The decimal point in the quotient belongs directly above the decimal point in the dividend.

Example

 

First, to make .425 a whole number, move the decimal point three places to the right: 425. Now move the decimal point three places to the right for 1.53: 1,530. The problem is now a simple long division problem.

 

Comparing Decimals

Comparing decimals is actually quite simple. Just line up the decimal points and then fill in zeros at the ends of the numbers until each one has an equal number of digits.

Example
Compare .5 and .005.

 

Line up decimal points.         .5

 

                                            .005

 

Add zeros.                            .500

 

                                            .005
Now, ignore the decimal point and consider, which is bigger: 500 or 5?
500 is definitely bigger than 5, so .5 is larger than .005.

Rounding Decimals

It is often inconvenient to work with very long decimals. Often it is much more convenient to have an approximation for a decimal that contains fewer digits than the entire decimal. In this case, we round decimals to a certain number of decimal places. There are numerous options for rounding:

To the nearest integer:           zero digits to the right of the decimal point
To the nearest tenth:           one digit to the right of the decimal point (tenths unit)
To the nearest hundredth:           two digits to the right of the decimal point (hundredths unit)

In order to round, we look at two digits of the decimal: the digit we are rounding to, and the digit to the immediate right. If the digit to the immediate right is less than 5, we leave the digit we are rounding to alone, and omit all the digits to the right of it. If the digit to the immediate right is 5 or greater, we increase the digit we are rounding by one, and omit all the digits to the right of it.

Example
Round to the nearest tenth and the nearest hundredth.
Dividing 3 by 7 gives us the repeating decimal .428571428571. … If we are rounding to the nearest tenth, we need to look at the digit in the tenths position (4) and the digit to the immediate right (2). Because 2 is less than 5, we leave the digit in the tenths position alone, and drop everything to the right of it. So, to the nearest tenth is .4.
To round to the nearest hundredth, we need to look at the digit in the hundredths position (2) and the digit to the immediate right (8). Because 8 is more than 5, we increase the digit in the hundredths position by 1, giving us 3, and drop everything to the right of it. So, to the nearest hundredth is .43.

Fractions

To work well with fractions, it is necessary to understand some basic concepts.

Simplifying Fractions

Rule:

• To simplify fractions, identify the greatest common factor (GCF) of the numerator and denominator and divide both the numerator and denominator by this number.
  Example
  
  The GCF of 63 and 72 is 9, so divide 63 and 72 each by 9 to simplify the fraction:
  

   

  

   

Adding and Subtracting Fractions

Rules:

To add or subtract fractions with the same denominator:

 

 

To add or subtract fractions with different denominators:

 

 

• To add or subtract fractions with like denominators, just add or subtract the numerators and keep the denominator.
  Example
  
• To add or subtract fractions with unlike denominators, first find the least common denominator or LCD. The LCD is the smallest number divisible by each of the denominators.

For example, for the denominators 8 and 12, 24 would be the LCD because 24 is the smallest number that is divisible by both 8 and 12: 8 × 3 = 24, and 12 × 2 = 24.

Using the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the appropriate factor to get the LCD, and then follow the directions for adding/subtracting fractions with like denominators.

Example

Multiplication of Fractions

Rule:

• Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators.
  Example
  

If any numerator and denominator have common factors, these may be simplified before multiplying. Divide the common multiples by a common factor. In the following example, 3 and 6 are both divided by 3 before multiplying.

Example

Dividing Fractions

Rule:

• Dividing fractions is equivalent to multiplying the dividend by the reciprocal of the divisor. When dividing fractions, simply multiply the dividend by the divisor's reciprocal to get the answer.
  Example
  (dividend) ÷ (divisor)
  

 

Determine the reciprocal of the divisor:

 

 

Multiply the dividend by the reciprocal of the divisor and simplify if necessary.

 

Comparing Fractions

Rules:

Sometimes it is necessary to compare the sizes of fractions. This is very simple when the fractions are familiar or when they have a common denominator.

Example

• If the fractions are not familiar and/or do not have a common denominator, there is a simple trick to remember. Multiply the numerator of the first fraction by the denominator of the second fraction. Write this answer under the first fraction. Then multiply the numerator of the second fraction by the denominator of the first one. Write this answer under the second fraction. Compare the two numbers. The larger number represents the larger fraction.
  Example
  Which is larger: or ?

 

Cross multiply.

 

7 × 9 = 63         4 × 11 = 44
63 > 44; therefore,

 

Compare and .

 

Cross multiply.
6 × 6 = 36         2 × 18 = 36
36 = 36; therefore,

Converting Decimals to Fractions

• To convert a nonrepeating decimal to a fraction, the digits of the decimal become the numerator of the fraction, and the denominator of the fraction is a power of 10 that contains that number of digits as zeros.
  Example
  Convert .125 to a fraction.
  The decimal .125 means 125 thousandths, so it is 125 parts of 1,000. An easy way to do this is to make 125 the numerator, and since there are three digits in the number 125, the denominator is 1 with three zeros, or 1,000.

 

 

Then we just need to reduce the fraction.
• When converting a repeating decimal to a fraction, the digits of the repeating pattern of the decimal become the numerator of the fraction, and the denominator of the fraction is the same number of 9s as digits.
  Example
  Convert to a fraction.

 

You may already recognize as . The repeating pattern, in this case 3, becomes our numerator. There is one digit in the pattern, so 9 is our denominator.

 

 

Example

 

Convert to a fraction.
The repeating pattern, in this case 36, becomes our numerator. There are two digits in the pattern, so 99 is our denominator.

Converting Fractions to Decimals

• To convert a fraction to a decimal, simply treat the fraction as a division problem.
  Example
  Convert to a decimal.
  
  So, is equal to .75.

Converting Mixed Numbers to and from Improper Fractions

Rule:

• A mixed number is number greater than 1 that is expressed as a whole number joined to a proper fraction. Examples of mixed numbers are . To convert from a mixed number to an improper fraction (a fraction where the numerator is greater than the denominator), multiply the whole number and the denominator and add the numerator. This becomes the new numerator. The new denominator is the same as the original.

Note: If the mixed number is negative, temporarily ignore the negative sign while performing the conversion, and just make sure you replace the negative sign when you're done.

Example
Convert to an improper fraction.

 

Using the conversion formula, .

 

Example
Convert to an improper fraction.

 

Temporarily ignore the negative sign and perform the conversion: .

 

The final answer includes the negative sign: .
• To convert from an improper fraction to a mixed number, simply treat the fraction like a division problem, and express the answer as a fraction rather than a decimal.
  Example
  Convert to a mixed number.
  Perform the division: 23 ÷ 7 = .

Percents

Percents are always "out of 100": 45% means 45 out of 100. Therefore, to write percents as decimals, move the decimal point two places to the left (to the hundredths place).

Here are some conversions you should be familiar with:

Absolute Value

The absolute value of a number is the distance of that number from zero. Distances are always represented by positive numbers, so the absolute value of any number is positive. Absolute value is represented by placing small vertical lines around the value: |x|.

Examples
The absolute value of seven: |7|.
The distance from seven to zero is seven, so |7| = 7.
The absolute value of negative three: |–3|.
The distance from negative three to zero is three, so |–3| = 3.

Exponents

Positive Exponents

A positive exponent indicates the number of times a base is used as a factor to attain a product.

Examples
Evaluate 2⁵.

 

In this example, 2 is the base and 5 is the exponent. Therefore, 2 should be used as a factor 5 times to attain a product:

 

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

Zero Exponent

Any nonzero number raised to the zero power equals 1.

Examples
5⁰ = 1       70⁰ = 1       29,874⁰ = 1

Negative Exponents

A base raised to a negative exponent is equivalent to the reciprocal of the base raised to the positive exponent (absolute value of the exponent).

Examples

Exponent Rules

• When multiplying identical bases, you add the exponents.
  Examples
  2² × 2⁴ × 2⁶ = 2¹²
  a² × a³ × a⁵ = a¹⁰
• When dividing identical bases, you subtract the exponents.
  Examples
  
• If a base raised to a power (in parentheses) is raised to another power, you multiply the exponents together.
  Examples
  (3²)⁷ = 3¹⁴ (g⁴)³ = g¹²

Perfect Squares

The number 5² is read "5 to the second power," or, more commonly, "5 squared." Perfect squares are numbers that are second powers of other numbers. Perfect squares are always zero or positive, because when you multiply a positive or a negative by itself, the result is always positive. The perfect squares are 0², 1², 2², 3²…

Perfect squares: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…

Perfect Cubes

The number 5³ is read "5 to the third power," or, more commonly, "5 cubed." (Powers higher than three have no special name.) Perfect cubes are numbers that are third powers of other numbers. Perfect cubes, unlike perfect squares, can be either positive or negative. This is because when a negative is multiplied by itself three times, the result is negative. The perfect cubes are 0³, 1³, 2³, 3³…

Perfect cubes: 0, 1, 8, 27, 64, 125…

• Note that 64 is both a perfect square and a perfect cube.

Square Roots

The square of a number is the product of the number and itself. For example, in the statement 3² = 3 × 3 = 9, the number 9 is the square of the number 3. If the process is reversed, the number 3 is the square root of the number 9. The symbol for square root is and is called a radical. The number inside of the radical is called the radicand.

Examples
52 = 25; therefore, = 5

 

Because 25 is the square of 5, it is also true that 5 is the square root of 25.

 

The square root of a number might not be a whole number. For example, the square root of 7 is 2.645751311.… It is not possible to find a whole number that can be multiplied by itself to equal 7. Square roots of nonperfect squares are irrational.

Cube Roots

The cube of a number is the product of the number and itself for a total of three times. For example, in the statement 23 = 2 × 2 × 2 = 8, the number 8 is the cube of the number 2. If the process is reversed, the number 2 is the cube root of the number 8. The symbol for cube root is the same as the square root symbol, except for a small 3: . It is read as "cube root." The number inside of the radical is still called the radicand, and the 3 is called the index. (In a square root, the index is not written, but it has an index of 2.)

Examples
53 = 125; therefore,

 

Like square roots, the cube root of a number might be not be a whole number. Cube roots of nonperfect cubes are irrational.

 

Probability

Probability is the numerical representation of the likelihood of an event occurring. Probability is always represented by a decimal or fraction between 0 and 1, 0 meaning that the event will never occur, and 1 meaning that the event will always occur. The higher the probability, the more likely the event is to occur.

A simple event is one action. Examples of simple events are: drawing one card from a deck, rolling one die, flipping one coin, or spinning a hand on a spinner once.

Simple Probability

The probability of an event occurring is defined as the number of desired outcomes divided by the total number of outcomes. The list of all outcomes is often called the sample space.

Examples

What is the probability of drawing a king from a standard deck of cards?

There are four kings in a standard deck of cards, so the number of desired outcomes is 4. There are 52 ways to pick a card from a standard deck of cards, so the total number of outcomes is 52. The probability of drawing a king from a standard deck of cards is . So, P(king) = .

Examples

What is the probability of getting an odd number on the roll of one die?

There are three odd numbers on a standard die: 1, 3, and 5. So, the number of desired outcomes is 3. There are six sides on a standard die, so there are 6 possible outcomes. The probability of rolling an odd number on a standard die is . So, P(odd) = .

Note: It is not necessary to reduce fractions when working with probability.

Probability of an Event Not Occurring

The sum of the probability of an event occurring and the probability of the event not occurring = 1. Therefore, if we know the probability of the event occurring, we can determine the probability of the event not occurring by subtracting from 1.

Examples
If the probability of rain tomorrow is 45%, what is the probability that it will not rain tomorrow?

 

45% = .45, and 1 – .45 = .55 or 55%. The probability that it will not rain is 55%.

 

Probability Involving the Word Or

Rule:

P(event A or event B) = P(event A) + P(event B) – P(overlap of event A and B)

When the word or appears in a simple probability problem, it signifies that you will be adding outcomes. For example, if we are interested in the probability of obtaining a king or a queen on a draw of a card, the number of desired outcomes is 8, because there are four kings and four queens in the deck. The probability of event A (drawing a king) is , and the probability of drawing a queen is . The overlap of event A and B would be any cards that are both a king and a queen at the same time, but there are no cards that are both a king and a queen at the same time. So the probability of obtaining a king or a queen is .

Examples
What is the probability of getting an even number or a multiple of 3 on the roll of a die?

 

The probability of getting an even number on the roll of a die is , because there are three even numbers (2, 4, 6) on a die and a total of 6 possible outcomes. The probability of getting a multiple of 3 is , because there are 2 multiples of three (3, 6) on a die. But because the outcome of rolling a 6 on the die is an overlap of both events, we must subtract from the result so we don't count it twice.

 

P(even or multiple of 3) = P(even) + P(multiple of 3) – P(overlap)

Compound Probability

A compound event is performing two or more simple events in succession. Drawing two cards from a deck, rolling three dice, flipping five coins, having four babies are all examples of compound events.

This can be done "with replacement" (probabilities do not change for each event) or "without replacement" (probabilities change for each event).

The probability of event A followed by event B occurring is P(A) × P(B). This is called the counting principle for probability.

Note: In mathematics, the word and usually signifies addition. In probability, however, and signifies multiplication and or signifies addition.

Examples
You have a jar filled with three red marbles, five green marbles, and two blue marbles. What is the probability of getting a red marble followed by a blue marble, with replacement?

 

"With replacement" in this case means that you will draw a marble, note its color, and then replace it back into the jar. This means that the probability of drawing a red marble does not change from one simple event to the next.

 

Note that there are a total of ten marbles in the jar, so the total number of outcomes is 10.

 

 

If the problem was changed to say "without replacement," that would mean you are drawing a marble, noting its color, but not returning it to the jar. This means that for the second event, you no longer have a total number of 10 outcomes; you have only 9, because you have taken one red marble out of the jar. In this case,