SESSION ONE

What are we going to cover this week?

Arithmetic:

Symbols & Operators

Summation and Subtraction

Product and Dividing

Equality and Inequality

Congruency and Proximity

The Geometrical Symbols

Some Terms

Some Basic Definitions

Number

Decimals

Digit

Place Value

Decimal Fraction

Repeating Decimals

Converting Decimals to Fractions

Converting Fractions to Decimals

Rules of Operation

PEMDAS Rule

Distributive Law

Associative Law

Commutative Law

Different Types of Numbers

Common Number Sets

o Real Numbers

Rational

Integers

Whole Numbers

Natural Numbers

Irrational

Even and Odd Numbers

Negative and Positive Numbers

Summation and Subtraction

Multiplying and Dividing

Absolute Values

Notes on Factors, Prime Numbers, and Prime Factorization

Factor

Product

Prime Number

Some Notes about Prime Numbers

Composite Number

Divisibility

o Parts of Division

o Divisibility Rules

Prime Factorization

Fraction

Adding and Subtracting Fractions

Multiplying and Dividing Fractions

Comparison of Fractions

Mix Numbers

Exponents & Roots

Exponents

Squares

Perfect Squares

Higher Order roots

Some General Rules

Simplifying roots

Adding roots

SYMBOLS & OPERATORS

SUMMATION AND SUBTRACTION

- Minus - Negative

+ Plus - Positive

PRODUCT AND DEVIDING

× or * or . Multiplied by

𝑎 ÷ 𝑏 𝑜𝑟 𝑎

𝑏 𝑜𝑟 𝑎: 𝑏 Divided by

EQUALITY, INEQUALITY

𝑎 = 𝑏 a is equal to b

𝑎 ≠ 𝑏 a is not equal to b

𝑎 > 𝑏 a is greater than b

𝑎 < 𝑏

a is less than b

𝑎 ≥ 𝑏 a is greater than or equal to

b

𝑎 ≤ 𝑏 a is less than or equal to b

CONGRUENCY AND PROXIMITY

≃ Is congruent to

≈ Is approximately equal to

THE GEOMETRICAL SYMBOLS

|| Is parallel to

∠B Angle B

⊾ Right angle

∠ABC Angle ABC

⊥ Is perpendicular to

SOME TERMS

Dozen 12

Billion 109

SOME BASIC DEFINITIONS

– Number

A number is a count or measurement that is really an idea in our minds.

We write or talk about numbers using numerals such as "4" or "four".

There are also special numbers (like π (Pi)) that can't be written exactly,

but are still numbers because we know the idea behind them.

– Decimals

A Decimal Number Based on the number 10

– Digit

A digit is a single symbol used to make numerals.

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are the ten digits we use in everyday numerals

(decimals).

123

– Place Value

When we write numbers, the position (or “place”) of each digit is

important.

So we can write:

447.2 = 4 × 100 + (4 × 10) + (7 × 1) + (2 × 0.1)

On the left of the decimal point as we move further left, every place gets

10 times bigger.

On the right side of the decimal point as we get further right, every place

gets 10 times smaller (one tenth as big).

digit digit digit

– Decimal Fraction

Decimal Fraction is a fraction where the denominator (the bottom number)

is a number such as 10, 100, 1000, etc. (a power of ten)

2.3 = 23

10

11.38104 →1138104

100000

– Repeating Decimal

Decimals that will repeat without end.

2

9= 0.2222 … = 0. 2̅

3

22= 0.13636 … = 0.136̅̅̅̅

– Convert Decimals to Fractions

Step 1: Write down the decimal divided by 1, like this: decimal 1.

Step 2: Multiply both top and bottom by 10 for every number after

the decimal point. (For example, if there are two numbers after the decimal point,

then use 100, if there are three then use 1000, etc.)

Step 3: Simplify (or reduce) the fraction.

– Convert Fractions to Decimals

Just divide the top of the fraction by the bottom, and read off the answer!

RULES OF OPERATION

– PEMDAS Rule

It is saying that you have to do calculations in parentheses (brackets) first, then

exponents next, and so on …

1. P = Parentheses (Brackets)

2. E = Exponents

3. M = Multiplication

D = Division

4. A = Addition

S = Subtraction

1 + 2 × 3

→ 𝑆𝑡𝑒𝑝 𝑂𝑛𝑒: Multiplication → 2 × 3 = 6

→ 𝑆𝑡𝑒𝑝 𝑇𝑤𝑜: Addition → 1 + 6 = 7

2 − 5 ÷ 1 + 9

→ 𝑆𝑡𝑒𝑝 𝑂𝑛𝑒: Division → 5 ÷ 1 = 5

→ 𝑆𝑡𝑒𝑝 𝑇𝑤𝑜: Addition → 2 − 5 + 9 = 6

6 + 9 × 32 − 5 − 1

→ 𝑆𝑡𝑒𝑝 𝑂𝑛𝑒: Exponents → 32 = 9

→ 𝑆𝑡𝑒𝑝 𝑇𝑤𝑜: Multiplication → 9 × 9 = 81

→ 𝑆𝑡𝑒𝑝 𝑇ℎ𝑟𝑒𝑒: Addition → 6 + 81 − 5 − 1 = 81

9 + 12 + 43 × 7 ÷ 2

→ 𝑆𝑡𝑒𝑝 𝑂𝑛𝑒: Exponents → 43 = 64 𝑎𝑛𝑑 12 = 1

→ 𝑆𝑡𝑒𝑝 𝑇𝑤𝑜: Multiplication → 64 × 7 = 448

→ 𝑆𝑡𝑒𝑝 𝑇ℎ𝑟𝑒𝑒: Division → 448 ÷ 2 = 224

→ 𝑆𝑡𝑒𝑝 𝑇ℎ𝑟𝑒𝑒: Addition → 9 + 1 + 224 = 234

'Please Excuse My Dear Aunt Sally'.

– Distributive Law

The Distributive Law says that multiplying a number by a group of numbers

added together is the same as doing each multiplication separately.

𝒂(𝒃 + 𝒄) = 𝒂𝒃 + 𝒂𝒄

– Associative Law

Within an expression containing two or more occurrences in a row of the same

associative operator, the order in which the operations are performed does not

matter as long as the sequence of the operands is not changed.

.𝒂 + (𝒃 + 𝒄) = (𝒂 + 𝒃) + 𝒄

(𝒂𝒃)𝒄 = 𝒂(𝒃𝒄)

– Commutative Law

In mathematics, either of two laws relating to number operations of addition

and multiplication, stated symbolically:

𝒂 + 𝒃 = 𝒃 + 𝒂

𝒂𝒃 = 𝒃𝒂

Example:

** It is a good idea to memorize these terms!

1. (𝒂 + 𝒃) × (𝒂 − 𝒃) = 𝒂 × (𝒂 − 𝒃) + 𝒃 × (𝒂 − 𝒃) = 𝒂𝟐 − 𝒂𝒃 +

𝒃𝒂 − 𝒃𝟐 = 𝒂𝟐 − 𝒃𝟐

2. (𝒂 + 𝒃)𝟐 = (𝒂 + 𝒃) × (𝒂 + 𝒃) = 𝒂 × (𝒂 + 𝒃) + 𝒃 × (𝒂 + 𝒃) =𝒂𝟐 + 𝒂𝒃 + 𝒃𝒂 + 𝒃𝟐 = 𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐

3. (𝒂 − 𝒃)𝟐 = (𝒂 − 𝒃) × (𝒂 − 𝒃) = 𝒂 × (𝒂 − 𝒃) + (−𝒃) × (𝒂 − 𝒃) =𝒂𝟐 − 𝒂𝒃 − 𝒃𝒂 + 𝒃𝟐 = 𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐

DIFFERENT TYPES OF NUMBERS

The set of all Whole Numbers:

{0,1,2,3,4,5, … }

The set of all Natural (Counting) Numbers:

{1,2,3,4,5, … } = 𝑁

The set of all Integers:

{… , −4, −3, −2, −1,0,1,2,3,4, … } = 𝑍

The set of Rational Numbers is a set of all numbers which can be expressed as:

𝑎 → 𝑁𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟

𝑏 → 𝐷𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

Where a and b are integers, and 𝑏 ≠ 0.

Examples: 2

3 ,

4

81 , 0.29 (=

29

100) , 0.1 (=

1

10) , 13 (=

13

1) , 0 (=

0

4), …

Note: 𝑎

0 = Not defined.

Note: Every integer is a rational number, since each integer n can be written in the

form 𝑛

1.

Irrational Numbers: is a real number that cannot be written as a simple fraction.

Can you have some examples?!

1.5 =15

10=

3

2→ 𝑅𝑎𝑡𝑖𝑜

𝑝𝑖 = 𝜋 = 3.14159 … = ?

?→ 𝑁𝑜 𝑅𝑎𝑡𝑖𝑜

Note: Any square root of any natural number that is not the square of a natural

number is irrational. √2 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑏𝑢𝑡 √144 𝑖𝑠 𝑛𝑜𝑡. 𝑊ℎ𝑦?

𝐵𝑒𝑐𝑎𝑢𝑠𝑒 √144 = 12 𝑎𝑛𝑑 12 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟.

Real numbers: All the rational and irrational numbers.

__________________________________________________________________

EVEN AND ODD NUMBERS

An even number is a number that can be divided into two equal groups.

An odd number is a number that cannot be divided into two equal groups.

Even numbers end in 2, 4, 6, 8 and 0 regardless of how many digits they have

(we know the number 5,917,624 is even because it ends in a 4!).

Odd numbers end in 1, 3, 5, 7, 9.

Some Tips:

even × even

6* 6 = 36

even

odd × odd

odd

even × odd

8*3 = 24

even

even ± even

even

odd ± odd

5-1 =4

even

even ± odd

4+1 = 5

odd

NEGATIVE AND POSITIVE NUMBERS

Positive Integers:

{1,2,3,4,5, … } = 𝑍+

Note: With positive numbers we typically do not write the algebraic sign +.

Negative Integers:

{−1, −2, −3, −4, −5, … } = 𝑍−

– Summation and Subtraction:

WE must give an Algebraic meaning to "adding" a negative number:

8 + (−2) =?

When we add a positive number, we get more. Therefore, when we "add" a

negative number, we must get less It means to subtract

8 + (−2) = 8 − 2 = 6

Here is the rule:

a + (−b) = a – b

Note: We use parentheses — a + (−b) — to separate the operation sign + from the

algebraic sign −. It would be bad form to write a + −b.

We can extend the rule for sums with more terms, here is a sum of four terms:

1 + (−2) + 3 + (−4) =?

According to the rule, we can remove the parentheses:

1 + (−2) + 3 + (−4) = 1 − 2 + 3 – 4

= 4 – 6

= – 2

– The rules for "adding" terms:

In algebra we speak of "adding," even though there are minus signs. With that

understanding, we can now state the rules for "adding" terms.

1) If the terms have the same sign, add their absolute values and keep that same

sign. Examples:

21 + 20 = 45

− 2 – 24 = − 26

− 3 + (− 22) = −3 – 22 = − 25

2) If the terms have opposite signs, subtract the smaller in absolute value from

the larger, and keep the sign of the larger. Examples:

2 + (−3) = −1

−2 + 3 = +1

Note: Again, in algebra we say that we "add" terms, even when there are

subtraction signs. And we call the terms themselves—and the answer—a

"sum." In other words, we always speak of a sum of terms.

Note: When working with singed integers, whenever you see one of the below

expressions, consider these rules:

– Multiplying and dividing:

Dividing is the inverse operation of multipling Simmilar Rules!

Examples:

2 × (+3) = 2 × 3 = 6

2 × (−31) = −62

−9 × 2 = −18

(−4) × (−1) = 4

6 ÷ (+3) = 2

2 ÷ (−2) = −1

−9 ÷ 2 = −4.5

(−32) ÷ (−2) = 16

ABSOLUTE VALUES

The absolute value of x is written as |x| and is defined as the positive value of x.

If x > 0 |x| = x | 4| = 4

If x < 0 |x| = -x |-4| = 4

The absolute value of x can also be thought of as the distance from 0 on a number

line to x.

Note: Distances are always positive.

Example: If |Y + 7| = 5 Y =?

We have two possible answers:

𝑦 + 7 > 0 → |y + 7| = y + 7 → y + 7 = 5 → y = 5 – 7 = −2

𝑦 + 7 < 0 → |y + 7| = −y − 7 → − y − 7 = 5 → −y = 5 + 7 → y = −12

The answer: 𝑦 = −2 or y = −12

Example: If |3Y+ 7| > 5, What is the range for Y?

3𝑦 + 7 > 0 → |3y + 7| = 3y + 7 → 3y + 7 > 5 → 3 y > −2 → y > −2

3

3𝑦 + 7 < 0 → |3y + 7| = −3y − 7 → −3y − 7 > 5 → −3y > 5 + 7

→ −3y > 12 → y < (12

−3) → 𝑦 < −4

The answer : 𝑦 > −2

3 or y < −4

NOTES ON FACTORS, PRIME NUMBERS, AND PRIME FACTORIZATION

– Factor

Factors are the numbers that multiply together to get another number.

All numbers have 1 and itself as factors.

– Product

A Product is the number produced by multiplying two factors.

– Prime Numbers

A number whose only factors are 1 and itself is a prime number. Prime numbers

have exactly two factors.

The smallest 168 prime numbers (all the prime numbers under 1000) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101,

103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193,197,

199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307,311,

313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421,431,

433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547,557,

563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659,661,

673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,809,

811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929,937,

941, 947, 953, 967, 971, 977, 983, 991, 997

– Some Notes about Prime Numbers:

There are infinitely many prime numbers.

The number 1 is not considered a prime.

2 is the smallest prime and the only even prime.

– Composite Number

A number with three or more factors is a composite number.

– Divisibility

o Parts of Division

Dividend = (Divisor × Quotient) + Reminder

11 = (2 × 5 )+ 1

Note: The values that the “reminder” can get cannot be equal or greater than

Divisor. → 0 ≤ Reminder < Divisor

o Divisibility Rules

All numbers are divisible by 1.

Any even number is divisible by 2.

A number is divisible by 2 if it ends with a 0, 2, 4, 6, or 8.

A number is divisible by 3 if the sum of its digits is divisible by 3.

A number is divisible by 4 if the last two digits (tens and ones) are

divisible by 4.

A number is divisible by 5 if it ends in a 5 or 0.

A number is divisible by 6 if it is also divisible by 2 and 3.

To test a number for divisibility by 7

Take the last digit in a number.

Double and subtract the last digit in your number from the rest

of the digits.

Repeat the process for larger numbers.

A number is divisible by 8 if the last 3 digits are divisible by 8.

A number is divisible by 9 if the sum of its digits are divisible by 9

A number is divisible by 10 if it ends in a zero.

– Prime Factorization

Every integer greater than 1 either is a prime number or can be uniquely

expressed as a product of factors that are prime numbers, or prime divisors Such

an expression is called a prime factorization.

To factor a number:

1. Write 1 and the number itself separated by some space.

2. Test the number for divisibility by 2. If it is, write 2 and the other number

inside the first two.

3. Continue testing and writing the factor pairs inside the previous pair.

4. When you reach the middle, you are finished.

Example: 36

{1 … 36}

{1, 2 … 18, 36}

{1, 2, 3 … 12 ,18 ,36}

{1, 2, 3,4 … 9, 12 ,18 ,36}

{1, 2, 3,4, 6, 9, 12 ,18 ,36}

So the factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.

To find the prime factorization of a number (with a factor tree):

1. Write the number to be prime factored at the top and draw two branches

below it.

1. Write its smallest prime factor on the left and circle it. If the number is

even, this will be 2. Write the companion factor on the right.

2. If the companion factor is composite, draw two branches below it and

repeat step 2 for this factor.

3. Continue repeating steps 2 and 3 until the companion factor on the right is

prime. Circle that prime also.

4. The prime factorization is the factors on the left and bottom of the tree that

are circled.

Example: 84

2 is the smallest prime factor of 84 along with 42

2 is the smallest prime factor of 42 along with 21

3 is the smallest prime factor of 21 along with 7

7 is prime so stop.

The prime factorization of 84 is: 2 × 2 × 3 × 7

FRACTRIONS

As we mentioned above, a fraction is a number of form 𝑎

𝑏 where a and b are integers

and b is not zero. The numerator is a and the denominator is b.

Such numbers are also named rational numbers.

Note: If both a and b are multiplied by the same nonzero integer, the resulting will

be equivalent to 𝑎

𝑏.

− 𝑎

𝑏= −

𝑎𝑐

𝑏𝑐

Example:

−2

3= −

4

6

Note: These terms are equal:

− 𝑎

𝑏=

−𝑎

𝑏=

𝑎

−𝑏

Example:

− 5

9=

−5

9=

5

−9

– Adding and Subtracting Fractions

To add or subtract two or more fractions with the same denominator, add or subtract

the numerator and keep the same denominator.

Example:

3

43+

5

43−

2

43=

6

43

To add or subtract two or more fractions with different denominators, first find a

common denominator. How?

It is a common multiple of the two denominators.

Then convert all fractions to equivalent fractions with the same denominator.

Finally, add the numerators and keep the common denominator.

Example:

3

4+

5

80−

2

20=

3 × 20

4 × 20+

5 × 1

80 × 1−

2 × 4

20 × 4=

60 + 5 − 8

80=

57

80

– Multiplying and Dividing Fractions

To multiply two or more fractions, multiply the numerators and multiply the

denominators.

Example:

(5

3) (

9

2) =

45

6

To divide one function by another, first invert the second fraction, then multiply

the first fraction by the inverted one.

Example:

(53)

(92)

= (5

3) ÷ (

9

2) = (

5

3) (

2

9) =

10

27

– Comparison of Fractions

When you are to say which fraction is grater or less than another one, it is wise

to make the denominators the same and then decide. This can be done by

multiplying the top and bottom of the fraction by the same number since this will

give a fraction with an equivalent value.

Example: Which fraction is bigger?

3

8 𝑜𝑟

5

12

3 × 3

8 × 3=

9

24 <

5 × 2

12 × 2=

10

24

– Mix Numbers

A mixed number has to parts

An integer part

A fraction part between 0 and 1

To convert a mixed number to a fraction, convert the integer part to an

equivalent fraction with the same denominator as the fraction, and then add it

the fraction part

EXPONENTS AND ROOTS

– Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself.

Example:

43 = 4 × 4 × 4

Note: There are two specially-named powers: "to the second power" is

generally pronounced as "squared", and "to the third power" is generally

pronounced as "cubed".

Note: When the negative numbers are raised to powers. The result may be

positive or negative.

If a negative number raised to an even number is always positive.

If a negative number raised to an odd power is always negative.

Example:

(−3)2 = 9

(−3)3 = −27

Note: look at this example:

−32 = −9

Exponents have a few rules that we can use for simplifying expressions:

Product Roles:

1. Whenever you multiply two terms with the same base, you can add the

exponents:

(𝑥𝑛)(𝑥𝑚) = 𝑥𝑛+𝑚

Example:

(67)(64) = 611

base

Exponent

2. Whenever you multiply two terms with the different bases but the same

exponents, you can use the following role: :

𝑥𝑛𝑦𝑛 = (𝑥. 𝑦)𝑛

Example:

5636 = (15)6

Quotient rules:

1. We can divide two powers with the same base by subtracting the

exponents.

(𝑥𝑛

𝑥𝑚) = 𝑥𝑛 𝑥−𝑚 = 𝑥𝑛−𝑚

Example:

53

57= 535−7 = 5−4

2. We can divide two powers with the different bases but the same

exponents:

(𝑥𝑛

𝑦𝑛) = (

𝑥

𝑦)

𝑛

Example:

63

33 = 23

Power Rules

1. Whenever you have an exponent expression that is raised to a power, you

can simplify by multiplying the outer power on the inner power:

(𝑥𝑛)𝑚 = 𝑥𝑛×𝑚

Example:

(𝑥𝑦2)3 = 𝑥3𝑦6

2.

𝑥𝑛𝑚= 𝑥(𝑛𝑚)

Example:

232= 2(32) = 29 = 512

Negative Exponents:

We can also have negative exponents, for all nonzero numbers x:

𝑥−𝑛 =1

𝑥𝑛

Example:

2−1 =1

21

5−2 =1

52

Zero rules

1. Anything (except 0) to the power zero is just "1"

𝑥0 = 1

(−3)1 = 1

31 = 1

2. For n>0:

0𝑛 = 0

Note: 00 is not defined.

One rules:

1. Anything to the power one is just equal to itself.

𝑥1 = 𝑥

(−3)1 = −3

31 = 3

2. When the base is 1, for every possible exponent, the answer is 1.

1𝑛 = 1

141 = 1

Note: For all nonzero numbers a:

𝑎 × (𝑎−1) = 1

– Roots

– Squares:

A square root of a nonnegative number n is a number r such that: 𝑟2 = 𝑛.

The symbol √𝑛 is used to denote the nonnegative square root of the nonnegative

number n.

Example:

32 = 9 → 3 𝑖𝑠 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 9 → 𝐴𝑛𝑜𝑡ℎ𝑒𝑟 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 9 𝑖𝑠 − 3

Note: All positive numbers have two square roots (- and +) because for

example:

42 = 16 𝑎𝑛𝑑 𝑎𝑙𝑠𝑜 (−4)2 = 16

√16 = ∓ 4

Note: The only square root of 0 is 0.

Note: square root of negative numbers, has no real solution. (Not used on

the GRE test)

Here are some useful rules:

Consider that x>0 and y>0

1. (√a)2

= a

2. √a2 = a

3. √a √b = √ab

4. √a

√b= √

a

b

– Perfect Squares:

Numbers with integers as their square roots: 4, 9, 16, etc.

To estimate square roots of numbers that aren’t perfect squares, just examine the

nearby perfect squares.

√16 < √17 < √25

4 < √17 < 5 𝑎𝑛𝑑 − 5 < √17 < −4

– Higher Order Roots

Higher order roots of a positive number n are defined similarly, For orders 3 and

4, the cube root of n is written as √𝑛3

, and forth root of n, written as √𝑛4

, represent

numbers such that when they are raised to the powers 3 and 4, respectively, the

results is n. these roots obey rules similar to those above.

Note: There are some notable differences between odd order and even order roots

(in the real number system)

For odd order roots, there are exactly one root for every number n, even

when n is negative. √8 3

= 2 8 has exactly 1 cube root.

For even order roots, there are exactly two roots for every positive

number n and no roots for any negative number n.

16 has 2 forth roots : 2 𝑎𝑛𝑑 − 2

-16 has no fourth root -16: negative

– Some General Roots:

Some general rules of roots:

– √(𝑥𝑛) 𝑚

= 𝑥𝑛

𝑚

Example:

√(26)2

= 23 = 8

– 𝑥1

𝑛 = √𝑥 𝑛

Example:

813

= √83

= 2

– Simplifying roots

When you work with roots in an equation, you often need to simplify them.

There are two methods: the quick, sort of intuitive method, and a slightly

longer method.

The quick method of simplification works only with some roots, like

√500

The quick method works for the square root of 500 because it’s easy to see a

large perfect square, 100, that goes into 500. Because 500 equals 100 times 5,

the 100 comes out as its square root, 10, leaving the 5 inside the square root.

The answer is thus:

100√5

It’s not as easy to find a large perfect square that goes into 504, so you’ve got

to use the longer method to solve it.

1. Break 504 down into a product of all of its prime factors.

√504 = √2 × 2 × 2 × 3 × 3 × 7

2. Identify each pair of numbers:

√504 = √2 × 2 × 2 × 3 × 3 × 7

3. For each pair you identify, take one number out.

√504 = 2 × 3 √2 × 7

4. Simplify

6√14

Example:

√−803

= √−1 × 2 × 2 × 2 × 2 × 5 3

−2 √2 × 5 3

= −2 √103

– Adding roots:

We can add or subtract radical expressions only when they have the same

radicand and when they have the same radical type such as square roots.

Example:

3√43

+ 5√43

− 7 √43

= √43

Sometimes we want to add or subtract radical expressions that does not have the

same radicand, in such situations, it is possible to simplify radical expressions,

and that may change the radicand. So the steps can be declared as:

1. Simplify each radical expression.

2. Add or subtract expressions with equal radicands.

3. To add or subtract like square roots, add or subtract the coefficients and keep

the like square root.

4. Sometimes when we have to add or subtract square roots that do not appear

to have like radicals, we find like radicals after simplifying the square roots

Example:

√48 − √75 = √2 × 2 × 2 × 2 × 3 − √5 × 5 × 3 = 4√3 − 5√3 = −√3