Math Definitions

Mathematics is one of the vast and most important concepts in education. From primary school mathematics is taught as a major subject introducing numbers, calculations and mathematical operations. In order to gain an in depth knowledge in mathematics, one needs to understand the various definitions or meaning of topics in Mathematics.

• Addition: The mathematical operation where two or more values are added is called addition. For example: She bought kids’ kitchen sets for Rs. 350 and stuffed cloth toys online for Rs.220. Therefore, she paid Rs. 350 + 220 = Rs. 570 while buying kids’ kitchen sets and stuffed cloth toys online.
• Subtraction: The mathematical operation where a part is subtracted from a whole is called subtraction. For example: Mahesh bought woodcraft construction kit online for Rs.1500. Seema contributed Rs. 580 in buying the woodcraft construction kit online. Therefore, Mahesh paid Rs. 1500 – 580 = Rs. 920 from his own pocket.
• Division: The mathematical operation where a value is distributed, it is called division. For example: Ram bought 8 apples which he needs to distribute among 4 kids equally. Therefore, 8 apples / 4 kids = 2 apples. Each kid gets 2 apples.
• Multiplication: The mathematical operation where a value is scaled by another value, it is called multiplication. For example: John earns $50 per day. Therefore, his earning per month is 50 * 30 = $1500.
• Algebra: Algebra is a branch in mathematics where one needs to find out the value of variables.  For example: 2X = 8. Therefore, X = 4.
• Fraction: Fraction is a part of a whole. For example: 7/9, 2/7 etc. are fractions.
• Proper Fraction: A proper fraction is a fraction where the numerator is smaller than the denominator. For example: 2/5, 3/9 etc.
• Improper Fraction: An improper fraction is a fraction where the denominator is greater than the numerator. For example: 5/7, 9/2 etc.
• Mixed Fraction: Mixed fraction is a combination of a whole number and a fraction. For example 3 ½ , 4 ½  etc.
These are some of the basic math definitions.

Free Algebra Equation Solver

Introduction:
Branch of mathematics deals with equations are called Algebra. In free Algebra equation solver we use letters like a, b,c.... y and z to denote numbers. Performing addition, subtraction, multiplication, division or extraction of roots and real numbers obtain free algebraic expressions. Symbols in free algebraic equation solver expression are called variables of expression. If two algebraic equation solver expressions are equated, we get free algebraic equations.


Simultaneous Equations


In free Algebra equation solver a set of linear equations having a common solution set is called a system of simultaneous linear equations. In earlier classes, have learnt different methods of solving simultaneous linear equations in two variables. linear equation in three unknowns x, y, z is a statement of equality of form ax + by + cz + d = 0 where a, b, c, d are real numbers with a ≠ 0, b ≠ 0 and c ≠ 0. For example 2x – 3y + 6z = 5 is a linear equation in 3 variables. Find values of three unknowns, given three linear equations in three unknown variables.

Solving given linear equations in x, y, z

Take any two equation solver.
First we are going to eliminate the z.
Now,we get 2 linear equations with x and y.
Solve them in common way learnt in previous cases.
By substitute x and y values in the equation,we get z value.
Finally we get x,y and z values.

I have recently faced lot of problem while learning System of Equations Solver, But thank to online resources of math which helped me to learn myself easily on net.

Practice Example


Example :

Solve:

x - 3y =12 ----------(1)
2x + 6y = 4 ----------(2)

SOLUTION:

(1) x 2 => 2x – 6y = 24
(2)      => 2x + 6y =   4   (+)
___________
4x + 0 = 28
___________

4x = 28

Divide by 4 on both sides,

(4x/4) = (28/4)

x = 7

Plug x = 7 in ( 1 ),

7 - 3y = 12

Subtracting 7 on both sides,

7 -3y - 7 = 12-7
-3y = 5

y = -(5/3)

Solution for this Algebra problem is x = 7 & y = -(5/3)

What Is Ratio In Math

Introduction for ratio in math:

In our daily life, by learning ratio and proportion many a times we compare two quantities of the same type. Thus, in convinced situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio. We denote ratio-using symbol ‘:’. If two ratios are equal, we state that they are in proportion and use the symbol ‘:’ or ‘=’ to equate the two ratios.

Ratio in math - Definition and example problems:


Definition of ratio:

The learning ratio is a comparison by division method. We compare the two quantities in terms of ‘how many times’. This comparison is known as the Ratio. We denote ratio-using symbol ‘:’

A : B

Ratio in math – Example problems:

Example 1:

If 5 pounds of grains cost $0.82, how much would 45 pounds of grains cost at the same rate?

Solution:

`(5 pounds) / ($0.82) ` = `(45 pounds) / (x)`   (Cross multiply)

(5 pounds) * (x) = (45 pounds) * ($0.82)

Divided by 5 pounds on both sides,

x = (9 pounds) * ($0.82)

x = $7.38

45 pounds of grains cost $7.38

Example 2:

The ratios of good scores to bad scores are 5 to 1. If there are 25 good scores, how many were bad?

Solution:

`(5 good scores) / (1 bad scores)` = `(25 good scores) / (x bad scores)`

5 * x = 25

X = `25 / 5`

X = 5

5 bad scores

Example 3:

Find the value for x,

`16 / 8 ` = `8 / x`

Solution:

`16 / 8` = `8 / x`

16 * x = 8 * 8

16x = 64                           (16 multiply 4 = 64)

x = `64 / 16`

x = 4

Example 4:

Find the value for x,

`8 / 13` = `8 / x`

Solution:

`8 / 13` = `8 / x`

8 * x = 8 * 13

8x = 104                           (13 multiply 8 = 104)

x = `104 / 8`

x = 13

Understanding Number and Operations is always challenging for me but thanks to all math help websites to help me out.

Ratio in math – Practice problems:


Practice problem 1:

Fine the value for x,

`12 / 8` = `6 / x`

Answer: x = 4

Practice problem 2:

Fine the value for x,

`14 / 7` = `6 / x`

Answer: x = 3

Practice problem 3:

In a backyard, the ratio of birds to squirrels is 8 to 2. If there are 16 squirrels in the backyard, how many birds are there?

Answer: 64 birds

Free Online Tutoring for Geometry

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free online tutoring for geometry - basic concepts

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conclusion

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Free Algebra Online

Introduction for free algebra online:
‘Free Algebra Online’ is a math tool which helps the students to get more knowledge in math. Also it supports to perform well in their exams. The student can come at any time though online and can get excellent help without any hesitation. Free algebra online covers the following basic operations such as addition, subtraction, multiplication and division. Also free algebra online cover the basic properties such as commutative, associative, identities and inverse. I like to share this free math solver algebra with you all through my article.


Most important term in free algebra online:


Free algebra online describes the terms such as variables, constant, coefficients, exponents, terms and expressions.

Variables

Algebraic variables are the alphabets where we are assigning the values. While solving the algebraic equation value of the variable will be changed. Widely used variables are x, y, z

Constant

Algebraic constants are the value whose values never changes while solving the algebraic equation. In 43y + 33, the value 33 is the constant.

Expressions

An algebraic Expression is the mixture form of variables, constant, coefficients, exponents, terms which are combined together by the following arithmetic operations such as Addition, subtraction, multiplication, and division. The example of an algebraic expression is given below

5y + 6

Term

Terms of the algebraic expression is used to form the algebraic expression by the arithmetic operations such as addition, subtraction, multiplication and division. In the following example 8n2 + 9n the terms 8n2, 9n are combined to form the algebraic expression 8n2 + 9n by the addition operation ( + )

Coefficient

The coefficient of an algebraic expression is the value present just before the terms. From the following example, 8n2 + 9n the coefficient of 8n2 is 8 and 9n is 9

Equations

An algebraic equation balances the numbers or expressions. Most probably algebraic equation is used for the value of the variable. The example of the equation is given below

9n +9 = 6


Order of the operation in Free algebra online


1. First, Reduce the algebraic expression whatever inside the parentheses.

2. Next, Reduce the exponents.

3. Next, Reduce the multiplication or division operations.

4. Finally, Reduce the addition or subtraction operations.

Having problem with Combinatorics Problems keep reading my upcoming posts, i will try to help you.

Examples for Free algebra online:


Example 1:

2(a-2)+4b-2(a-b-2)+10

Solution:

=  2(a-2)+4b-2(a-b-2)+10

= 2a–4+4b–2a+2b+4+10

= 2a–2a+4b+2b-4+4+10

= 6b+10

= 3 ( 3b – 5)

Example 2:

5x - 15 = 4x - 30

Solution:

5x - 15 = 4x - 30

5x – 15 + 15 = 4x – 30 + 15 (Add 15 on both sides)

5x =4x -15

5x – 4x =4x -4x - 15 (Add -4x on both sides)

x = -15

Angle of Depression Problems

Introduction to Angle Of Depression problems:

The word “depression” means “fall” or “drop”.

Angle of depression problem is the angle which is between the horizontal and the line of sight to an object beneath the horizontal. The angle of Depression is similar to angle of elevation.

The angle of depression problem is below the horizontal that an observer must look to see an object that is lower than the observer.The angle of depression is similar to the angle of elevation (this assumes the object is close to the observer so that the horizontals for the observer and the object are parallel; this would not be the case for an astronaut in orbit around the earth observing an object on the ground).

Please express your views of this topic What is the Angle of Depression by commenting on blog.

Angle of depression problem:1

A man who is 2 meter tall stands on horizontal ground 30 meter from a tree. The angle of elevation from the top of the tree from his eyes is 28˚. Find the height of the tree.

Angle of depression problem solution:

Let the height of the tree be h.
Tan 28o=(h-2)/30
h – 2 = 30 tan 28˚
h = (30 ´ 0.5317) + 2 ← tan 28˚ = 0.5317
= 17.951
The height of the tree is approximately 17.95 meter.

Angle of depression problem:2

An Bird is flying at an altitude of 1000 m. The angle of depression from the bird of a tree on the ground is measured as 15°. What is the distance from the bird to the tree?
Choices:
A. 3,864 m
B. 3,732 m
C. 1,035 m
D. 1,000 m
Angle of depression problem solution:
Step 1: Let x be the distance from the bird  to the tree.
Step 2: Here the horizontal lines are parallel; the angles of elevation is equal to angle of depression Step 3: sin 150=1000/x
Step 4: x=1000/sin 150
Step 5: x = 3,864 m
Step 6: the distance from the bird to the tree is 3,864 m.

Learn Algebra 2

Definition of algebra:

Algebra is the division of arithmetic to the study of the regulations of operations and dealings, and the concept arising from them, together with all possible algebraic sub topics. Algebra is easy to learn and to solve the problems when the concept is understood very well.

An algebraic term represent a scale, what is ready on one side of the scale with a number is also ready to the other side of the range. The numbers are the constants. Algebra consist of real numbers, complex numbers, matrices, vectors etc.

Free answers to algebra problems:

Q1:  A person crosses a 500 m long street in 7 minutes. What is his speed in km per hour

Solution:

Given,

Distance = 500m

Time = 7minutes

Formula:

Speed = distance / time

Speed = 500 / (7*60)

Speed = 1.2m/sec

Converting m/hr to km/hr, we get

=1.2*18/7

=3.08km/hr

Answer:

The final answer is 3.08km/hr

Q2: The ratio between the speeds of two trains is 5 : 6. If the second bus runs 300 km/s in 2 hours, then find the speed of the first bus in km/hr.

Solution:

Given ,

The speed of two bus be 5x and 6x

Distance = 300km

Time = 2hours

Speed formula:

Speed = distance/ time

First take 5x,

5x = 300 / 2

= 150

X = 150/5

= 30 km/hr

Answer:

Speed of first bus=30*6=180km/hr

Q3: A man complete a journey in 8 hours. He appointments first half of the journey at the rate of 20 km/hr and second half at the rate of 22 km/hr. Find the total journey in km.

Solution:

(½) x / 20+ (1/2) x / 22 = 8

After simplifying, we get

X / 20 + x / 22 = 8

Take lcm for 20 and 22, we get

Lcm is 220

21x = 220*8

21x = 1760

Divide by 21 on both the sides, we get

X = 1760 / 21

= 83.8km

Answer:

The final answer is 83.8 km. Understanding Inverse Matrix Calculator is always challenging for me but thanks to all math help websites to help me out.


Practice questions for free algebra:

Question 1:

The rectangle of a two digit integer is divided by half the number. After 36 is added to the measure, this sum is then divided by 2. The digit of the resulting numbers are the equal as the new number, but they are in reverse order. The tenth place of the number is equal to twice the difference between its digits. What is the number?

Answer:- 46

Question 2:

In one hour, a ship goes 11 miles per hour along the stream and 5 miles per hour next to the river. The rate of the ship in still stream (in miles per hour) is:

Answer: - 8 miles per hour

Never Happen Events

Introduction to never happen events :

Never happen events are events that we come across  in probability.

We know that probablity ranges between 0 and 1

Sure events have the probability of one.

But there are certain events that cannot occur at all.

These events are termed as never happen events. I like to share this Exclusive Events with you all through my article.

Neven happen events examples:-

These never happen events are events that are impossible to occur.

Let me give you a few examples.

1)The sun  rises at  night.

This is impossible.  So we call it a never happen event.

2)The moon does not wane or wax.

This is also a never happen event.

When anybody says "I saw a cow studying in a school" , you would say "It is impossible"

That is a never happen event.

3) A man living forever is a never happen event.

4) Electing 2 Presidents for a country is a never happen event.

5) A year having more than  365 or 366(leap year)  days is a never happen event.

6)Having 8 days a week is a never happen event.

7)Having 15 months in a year is a never happen event.

Understanding Intersecting Lines Definition is always challenging for me but thanks to all math help websites to help me out.


Never happen events in mathamatic:-


Probability of drawing an animal card from a pack of  52 cards is a never happen event.

Getting a  pair of heads in a toss of coin is a never happen event.

Getting a number greater than 6 in a throw of a die is a never happen event.

Getting triple heads from a throw of 2 coins is a  never happen event.

Getting more than 36 possible outcomes from a throw of 2 dices is a never happen event.

Getting a total of 13 when 2 dices are thrown up is a never happen event, because the maximum total you can get is

6 from the fist dice and 6 from the second dice that total up to 12.

Drawing 14 spade cards form a pack of 52 cards is a never happen event.(There are only 13 spades in a pack of card)

Getting a  prime number that is even, other than 2 is a never happen event( because 2 is the only even prime number)

Getting a prime number less than 2 is a never happen event, because prime starts from the number 2 onwards.

Getting an odd number when you add two odd numbers is a never happen event, because any 2 odd numbers when

added gives an even number.  For example 9 +11=20(even number)   and 49 + 51 =100(even number)

The probability of a event exceeding 1 is also a never happen event.

If the probability is said to be 1.5 then it is unlikely to occur. So we call it a never happen event

A student scoring 110 out of 100 is a never happen event.  The probability works out to be 1.1 which is impossible.

All these examples shows that if an event is unlikely to occur, it is improbable. We term it as never happen events.

Algebra Solver Free

Introduction to algebra solver free:

Algebra is a branch of mathematics. It contains the all operations between the numbers. By using these operations we find the values.The operations are addition, subtraction, multiplication and division is taken between numbers and fractions and many more.

Ex:        1) (60-36)/(3*4)

24/12 =2

2) When 20/24 is reduced to lowest term.

20/24 =10/12 =5/6

3) Add and write the answer

4/3 +1/2 + 5/15 = (40+15+ 10)/30 = 65/30 = 13/6

Operations and their properties:- In mathematics mainly we have four fundamental operations (+ , - *, / ) on numbers.

1) 3+4 =7

3 Ø 4 Where Ø represents addition

2) 7-2 =5

7Ø2 Where Ø represents subtraction

3) 5*2=10

5Ø2 Where Ø represents multiplication

4) 25 /5 =5

25Ø5 Where Ø represents division

Thus the symbol Ø may be used to represent any of the four fundamental operations on given set of numbers P.


Properties in algebra solver free


1)Closure property:- If a, b belongs to real numbers and aØb is also belongs to P we say P is closed under the operation.

2)Commutative property:- If a, b belongs to P, Where P is the set of numbers and aØb =bØa, then P is commutative with respect to Ø

3)Associative property:- If a Ø "(b Ø c) = (a Ø b) Ø c, then we say P is Associative with respect to Ø

4)Identity property:- If a, b belongs to P and there exist and element  e belongs to P and a Ø e = e Ø a = a for all "a" belongs to P, then e is called Identity element in P.

In the set of rational numbers, the value for e is  zero if Ø represents addition and 1 if Ø represents multiplication.

5)Inverse:- If a belongs to P and a'  belongs to P and a Ø a' = e = a' Ø a then we say that  a' is the inverse of a with respect to Ø in P. If Z and Q the inverse of a is -a in the operation is addition and also for set Q and R the inverse of a is 1/a in the operation on multiplication. that is a' = 1/a, a is not equal to zero.

6)Distributive law:-If two binary operations addition and multiplication are defined over the set P, then the distributive law is  a*(b+c) = (a*b)+ (a*c) this is the multiplication is distributive over addition.

Decimal representation of rational numbers:- Rational numbers can be expressed in the decimal form by dividing the numerator by denominator.

Ex:- 1/2 = 0.5, 1/8=0.125, 1/4=0.25

The denominator of any decimal fraction is 10 or power of 10.

Ex:-0.5 = 5/10, 0.25=25/100, 0.125 =125/1000

If a fraction, in its lowest terms, has no other prime factor except 2 and 5 it can be expressed as a terminating decimal.

Non terminating decimals:- If a rational number in its lowest form has prime factor other than 2 and 5 during the process of division we get a digit or group of digits are repeated in the same order such decimal fractions are called non terminating decimals.

Ex:- 1/3 = 0.3333-----------

7/15 =0.4666----------

Order of operations:-(PEDMAS RULE):-The operations inside the brackets that is (   ) or [   ] are done first. Then any operation involving exponents then do all multiplying and divisions from left ot right finally do all additions and subtractions from left to right.


Solve the problems using algebra solver free:


Equation with one variable:-The simple equation having the only one variable that is x and having constant term. It is always does not contains logarithmic and exponents.

Ex 1:solve the equation 3x+4 =5

Sol:                        given 3x+4 = 5

`rArr` 3x = 5-4

`rArr` 3x=1

`rArr` x  = 1/3.

2)Solve the system of equations having two variables 4x+2y =14 and 2x-y =1

Sol:     Given equations are 4x+2y=14

2x-y =1   this equation multiple with 2  we have

4x-2y =2

4x+2y =14

4x-2y =2

____________

8x =16

` x = 16/8 = 2`

Therefore x =2

The value of x is substituted in the 2x-y = 1

2(2) -y = 1

4-y = 1

4-1 =y

y =3

The values of x =2 and y =3

live algebra help

Introduction to live algebra help:
Algebra help is an important part in mathematics here we use alphabets, symbols which we use to represent the number system. Algebra include actual numbers, complex numbers, matrices, vectors etc.Processing with unknown standards is called variables, unlike math which is based on completely identified number values. Students can have live help in algebra through online. Through live algebra help students get clear idea about the topic. Live help through online has many advantages then class teaching. In online live help students gets one to one help. I like to share this math help for algebra 2 with you all through my article.


Live algebra help:


Variables:

The lowercase alphabets are used as a variable.

Exponents:

Exponents are identified the variables using the ^ (power) symbol.

For example, to express x3, enter x^3.

Note: exponents should be positive numbers, no negative numbers, decimals, or variables. Exponent cannot currently be placed on numbers, brackets, or parentheses.

Parentheses and Brackets:

Parentheses ( ) and brackets [ ] it is used to group a particular set of variable and their exponent terms as in a standard equation or expression.

Multiplication, Addition, and Subtraction:

In addition and subtraction, use the symbols like + and - symbols. For multiplication, you can use the * symbol.

A * symbol is not needed when multiplying a number by a variable. For 3 * x can also be identified as 3x. Similarly, 4 * (x + 3) can also be entered as 4(x + 3); 4x * (12) can be entered as 4x(12). The * is also optional when you can multiply with parentheses.

Example: (x + 2)(x - 2).

Order of Operations:

The standard order of operations shows nearly all algebra notations Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. The only identification is that divisions of particular variable denote to the attempts to use the symbol will be the result in an error Understanding Simplify Expressions Calculator is always challenging for me but thanks to all math help websites to help me out.


Live algebra help problems:


1) Define the expression 2x.

2x is an example of single term and It has factor of 2 and x.

The 2 is called the coefficient of the expression and the x is a variable.

2) 2x + 6y expression .

First term: 2x, has factors 2 and x

Second term: 6y, has factors 6 and y

The 2 and 6 are called the coefficients of the expressions.

3) Simplify: 7x + 2y +x -11 .

Solution:

The answer for given question is 7x + x + 2y -11

7x + x = 6x here added the x terms.

The answer 8x + 2y -11.

Definition of Complement

Introduction to definition of complement:

The complement of set A, indicated by A’, is the set of every elements in the universal set that are not in A. The numeral of elements of A and the numeral of elements of A’ makes up the whole number of elements in U. Let us see about the topic definition of complement are given below examples. Having problem with Binomial Expansion Calculator keep reading my upcoming posts, i will try to help you

Example Problems for Definition of Complement Set

Let us see about the topic definition of complement set are given some example problems with solution are shown in below,

Example problem 1:- Using definition of complement

Let U = {x: x is an integer, –6 = x = 7}, P = {–4, –2, 1, 4, 3, 6, 7} and

Q’ = {–3, –2, –1, 2, 3}. List the elements of set P ’

Solution:

Given:

List the elements of set P ’

First, list out the members of U.

U = {-6,-5,–4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6,7}

P’ = {–5,-3,–1, 2, 5, 6} ? in U but not in P

Example problem 2:- Using definition of complement

Let U = {x: x is an integer, –6 = x = 7}, P = {–4, –2, 1, 4, 3, 6, 7} and

Q’ = {–3, –2, –1, 2, 3}. Find n (Q)

Solution:

Given:

Step 1: to calculate n (Q)

U = {-6,-5,–4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6,7} and P’ = {–5,-3,–1, 2, 5, 6}

n ( U ) = 14, n(Q ’ ) = 6

Please express your views of this topic prime numbers 10000 by commenting on blog.

Example Problem 3:- Using Definition of Complement

Let U = {x: x is an integer, –6 = x = 7}, P = {–4, –2, 1, 4, 3, 6, 7} and

Q’ = {–3, –2, –1, 2, 3}. Find n (Q)

Solution:

Given:

Step 1: to calculate n (Q)

U = {-6,-5,–4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6,7} and P’ = {–5,-3,–1, 2, 5, 6}

n ( U ) = 14, n(Q ’ ) = 6

Use the method:

n (Q) + n (Q’ ) = n ( U )

n (Q) = n (U) – n (Q ’ ) = 14 – 6 = 8

Example problem 4:- Using definition of complement

U = set of positive integers less than 7 x = {7, 4, 2, 6, 1} and y = (1, 2, 3, 5}

Find the (X n Y) ‘

Solution:

Given:

Step: 1 we can determine the values of the (X n Y) ‘= {7, 4, 6, 5}

Step: 2 Another method we can calculate (X n Y) = {1, 2}

Step: 3 (X n Y) ‘= {7, 4, 6, 5}

Step by Step Math Help

Introduction to step by step math help

Step by step help in math is essential to understand and solve the problem.
The first step is to read  the problem.
The second step is to understand the problem
The third step is to interpret  the problem in parts
The fourth step is to  find the formula to solve
The fifth step is to apply the formula and solve
The final step is to check our solution.

Step by Step Math Help -examples

Let us solve this algebra problem step by step

Problem 1 :
Five added to twice a  number gives  21.  Find the number.


Let us analyse the problem step by step


Let the number be x
Five added                         +5
to twice a number :              2x
Now write  both together as 2x +5
gives                                  =
21                                     21


Now let us write it in the proper form  2x + 5 = 21


Let us solve the problem
Transposing +5 to the other side we get  2x = 21- 5
2x = 16
Dividing by 2 we get                                x = 8


Now let us check 2(8) + 5 = 16 + 5 =21
Hence the problem is verified.

Step by Step Math Help-practice Problems
Let us do a problem in arithmetic

Problem 2 :
Multiply   137 x 137 x 137 - 67 x 67 x 67
--------------------------------------------
137 x 137 + 137x 67 + 67 x67
On close examination of the problem we find that it obeys a certain algebra formula
The formula is a3 - b3 = (a-b) ( a2+ ab + b2)
Dividing both sides by (a2+ab+b2)  we get  a3 - b3  =  (a-b)
-------------
(a2+ab+b2)
In this problem  137x137x137 = a3
67x67x67    = b3
and  137x137 = a2      137x67 = ab and 67x67 = b2
Hence we write the problem as        (137)3  - (67)3               (137 - 67)    =  60
----------------------------------   =
(137)2+ (137)(67) + (67)2
Hence the answer to this problem  is 60
We analysed this problem and did it step by step
Problem 3 :
Find the value of( 243)2/5
First step ;-  We must first find   the 5th root of 243 first.
second step  243 = 3x3x3x3x3 = 3 5
Now write the problem (243)2/5    as  (3 5) 2/5
Next write  the  problem as (3)(5x 2/5) =  (3)2 =   9
Problem 4 :
Problem on gain %
If the cost price of 6 toffees equals the selling price of 4 toffees, fin the gain %
Step 1  Le us assume the cost price of 1 toffee  to be $1
Then cost price of 4 toffees =  $4
Next step  Selling price of  4 toffees = cost price of 6 toffees = $6
Now we know CP of 4 toffees  = $4  and selling price  of 4  toffees  = $ 6
Hence gain on 4 toffees = $6 - $4 = $2
Gain%  =( gain x100)%          (    2 x100 ) %
--------                      =    -----                  =    50%
CP                                4