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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