Class 8

Students can go through AP 8th Class Maths Notes Chapter 16 Playing with Numbers to understand and remember the concepts easily.

Class 8 Maths Chapter 16 Notes Playing with Numbers

→ Numbers can be written in general form. Thus, a two digit number ab will be written as ab = 10a + b.

→ The general form of numbers are helpful in solving puzzles or number games.

→ The reasons for the divisibility of numbers by 10, 5, 2, 9 or 3 can be given when numbers are written in general form.

→ We are familiar with different number systems like Natural numbers, Whole numbers, Rational numbers, etc. and the properties of those number system.

→ To improve our interest in mathematics we can think of different concepts like addition, division, multiplication etc. in different methods.

→ This is we say “Play with Numbers” which very interesting and thought provoking.

→ A number we write usually (like 27, 52, 108, 260868 etc.) are in usual form.

→ We can express them in generalised form also. Generalised form is nothing but expressing the digits with their number place system.
For
Ex : 23 → im = 2 is in tens place and 3 is in units place.
So 23 can be expressed as (2 × 10) + (3 × 1)
and 57 can be expressed as (5 × 10) + (7 × 1)
and 60 can be expressed as (6 × 10) + (0 × 1)
98 can be expressed as (9 × 10) + (8 × 1)
123 can be expressed as (1 × 100) + (2 × 10) + (3xl)
548 can be expressed as (5 × 100) + (4 × 10) + (8 × 1)
Thus (2 × 10) + (3 × 1),
(5 × 10) + (7 × 1),
(9 × 10) + (8 × 1) all these are in generalised forms.

Students can go through AP 8th Class Maths Notes Chapter 15 Introduction to Graphs to understand and remember the concepts easily.

Class 8 Maths Chapter 15 Notes Introduction to Graphs

→ Graphical presentation of data is easier to understand.

→

1. A bar graph is used to show comparison among categories.
2. A pie graph is used to compare parts of a whole.
3. A Histogram is a bar graph that shows data in intervals.

→ A line graph displays data that changes continuously over periods of time.

→ Aline graph which is a whole unbroken line is called a linear graph.

→ For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.

→ The relation between dependent vari-able and independent variable is shown through a graph.

→ Graphs : Graphs are visual representations of collected data.

→ What is the purpose of graph ?
Purpose of graph is to show numerical facts in visual form so that they can be understood quickly, easily and clearly.

The following figure shows marks of a child in SA I, SA II, SA III, which can be represented by bar graph.

→ Pie – graph or Circle graph: Pie graph is another kind of graph to represent data in sectors.

Histogram : Histogram is a bar graph that expresses data in intervals, it has adjacent bars over the intervals
The following figure illustrates the distribution of the data given below table.

Weight (kg)	40 – 45	45 – 50	50 – 55	55 – 60	60 – 65
Number of persons	4	12	13	5	5

→ If gaps are there in given intervals, then there will be gaps in bars of Histogram. Otherwise there wont be any graps.

→ Line graph : A line graph displays data that changes continuously over periods of time.
Ex : distance – speed ; V – T (velocity time graph) t – t (time, temperature) graph.

In this if time is taken on horizontal line (X – axis the other one temperature / velocity etc., will be taken on Vertical line (called Y – axis)

Students can go through AP 8th Class Maths Notes Chapter 14 Factorization to understand and remember the concepts easily.

Class 8 Maths Chapter 14 Notes Factorization

→ When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic vari-ables or algebraic expressions.

→ An irreducible factor is a factor which cannot be expressed further as a product of factors.

→ A system atic way of factorising an expression is the common factor method. It consists of three steps : (i) Write each term of the expression as a product of irreducible factors (ii) Look for and separate the common factors and (iii) Combine the remaining factors in each term in accordance with the distributive law.

→ Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.

→ In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.

→ A number of expressions’to be factorised are of the form or can be put into the form : a² + 2ab + b², a² – 2ab + b², a² – b² and x² + (a + b) + ab. These expressions Can be easily factorised using Identities I, II, III and IV.
I. a² + 2ab + b² = (a + b)²
II. a² – 2ab + b² = (a – b)²
III. a² – b² = (a + b) (a – b)
IV. x² + (a + b) x + ab = (x + a) (x + b)

→ In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab. Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.

→ We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.

→ In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.

→ In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.

→ In the case of divisions of algebraic expressions that we studied in this chapter, we have
Dividend = Divisor × Quotient.
In general, however, the relation is Dividend = Divisor × Quotient + Remainder
Thus, we have considered in the pre¬sent chapter only those divisions in which the remainder is zero.

→ Factor : An integer is solid to be a factor, if it divides the other number without any remainder.
Ex: 4 divides 36 ; hence 4 is a factor.
9 divides 36 ; hence 9 is a factor.
6 divides 36 ; hence 6 is a factor.
12 divides 36; hence 12 is a factor.
18 divides 36; hence 18 is a factor.
36 divides 36; hence 36 is a factor.
Thus 1, 2, 3, 4, 6, 9, 12,18, 36 are factors of 36.
Ex : Write factors of 48
48 = 1 × 48, 2 × 24, 3 × 16,4 × 12, 6 × 8 Hence 1, 2, 3, 4, (3, 8, 12, 16, 24 and 48 are factors of 48.

→ Factors of Algebraic Expression:
Expression 7x²y + 8xy²
Factors of 7x²y = 7 × (x) × (x) × (y)
and factors of 8xy² = [2 × 2 × 2 × (x) × (y) × (y)]

Factorisation : Writing the given as product of its factors is called factorisation. (Factors should be in its irreducible form)
Ex : L Factorise 2x + 6
Factors of 2x = 2, x = 2(x)
Factorising 6 = 2 × 3
then 2x + 6 = 2(x) + 2(3)
here ‘2’ is common factor.
then 2x + 6 can be expressed as 2(x) + 2(3)
= 2(x + 3) (picking the common factor 2)
∴ 2x + 6 = 2(x + 3)
Hence the factor of 2x + 6 are (2) and (x + 3) which are irreducible.
Ex : II. 8xy + 10x²
Factorising 8xy = 2 × 2 × 2 × (x) × (y)
Factorising 10x² = 5 × 2 (x)(x)
∴ 8xy + 10x² = 2 × 2 × 2 × (x) × (y) + 5 × 2 (x)(x)
Picking common factors out 2, x = 2x(4y + 5x)
This is the required factor form

Students can go through AP 8th Class Maths Notes Chapter 13 Direct and Indirect Proportions to understand and remember the concepts easily.

Class 8 Maths Chapter 13 Notes Direct and Indirect Proportions

→ Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if \(\frac{x}{y}\) = k [k is a positive number], then x and y are said to vary directly. In such a case if y₁, y₂ are the values of y corresponding to the values x₁, x₂ of x respectively then \(\frac{\mathrm{x}_1}{\mathrm{y}_1}\) = \(\frac{\mathrm{x}_1}{\mathrm{y}_1}\).

→ Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. Tfiat is, if xy = k, then x and y are said to vary inversely. In this case if y₁, y₂ are the values of y corresponding to the values
x₁, x₂ of x respectively then x₁, y₁ = x₂, y₂ or \(\frac{\mathrm{x}_1}{\mathrm{x}_2}\) = \(\frac{\mathrm{y}_2}{\mathrm{y}_1}\)

Concepts of Variation :
→ Constants: 5, 10, 0, – 8. – 7,…. all these are constants values of these are always fixed irrespect of time, place.

→ Variables : temperature, wind speed, rainfall, price of a share, price of oil, etc. values of these depend on some other parameters hence they change from time to time, (or) place to place. So we call them variables.
The way they change is called variation In the middle classes we learn two types of variations. They are :

1. Direct variation (or) Direct Proportion
2. Indirect (or) inverse variation (or) Indirect proportion.
   In higher classes we learn some other variations like combined variation, partial variation, etc ….

→ Direct variation : Perimeter of a circle depends on its radius in a fixed way. So, Perimeter (c) is directly proportional to its radius (r)
Price of petrol pumped into our tank is directly proportional to price of 1 litre petrol.

→ Price of painting is directly proportional to the area painted …. etc are some examples.

→ Direct Proportion: If the change in one variable is in same rate of change in other variable then they (two variables) are in direct proportion.
If x, y are two variables in direct proportion then we write it as x ∝ y (read as ‘x’ is directly proportional to ‘y’)
∝ – (proportional) is the symbol we use.
When x ∝ y ; then their ratio is always fixed and it is called proportional constant.
If x ∝ y ⇒ \(\frac{x}{y}\) = ky (proportional constant) or (rate of change) or x = ky
IMP : If the ratio is not same everytime then we cannot say they are in direct proportion.

Students can go through AP 8th Class Maths Notes Chapter 12 Exponents and Powers to understand and remember the concepts easily.

Class 8 Maths Chapter 12 Notes Exponents and Powers

→ Numbers with negative exponents obey the following laws of exponents.
a) a^m × aⁿ = a^{m – n}
b) a^m ÷ aⁿ = a^{m – n}
c) (a^m)ⁿ = a^mn
d) a^m × b^m = (ab)^m
e) a⁰ = 0
f) \(\frac{a^m}{b^m}\) = (\(\frac{a}{b}\))^m

→ Very small numbers can be expressed in standard form using negative exponents.

→ a^m, 2⁵, 5^-4, 10¹⁰⁹, ………… exponent forms.

→ Find the value of 2⁵ ; 2⁵ = 2 × 2 × 2 × 2 × 2 = 32 (5 times multiplication)

→ a^m in this form
‘a’ is called base ; ‘m’ is called exponent.
10^-3 = \(\frac{1}{1000}\) or \(\frac{1}{10^3}\)

→ For a now zero integer a, a^-m = \(\frac{1}{\mathrm{a}^{\mathrm{m}}}\) where ‘m’ is a positive integer.

Students can go through AP 8th Class Maths Notes Chapter 11 Mensuration to understand and remember the concepts easily.

Class 8 Maths Chapter 11 Notes Mensuration

→ Area of

1. a trapezium = half of the sum of the lengths of parallel sides × perpendicular distance between them.
2. a rhombus = half the product of its diagonals.

→ Surface area of a solid is the sum of the areas of its faces.

→ Surface area of
a cuboid = 2(lb + bh + hl)
a cube = 6l²
a cylinder = 2πr(r + h)

→ Amount of region occupied by a solid is called its volume.

→ Volume of
a cuboid = l × b × h
a cube = l³
a cylinder = πr²h

→ i) 1 cm³ = 1 mL
ii) 1L = 1000 cm³
iii) 1 m³ = 1000000 cm³
= 1000L

→ Perimeter, and area can be calculated for closed figures only.

→ Perimeter is the distance around the boundary of given closed figure.

→ Area is the region covered by the boundary of given closed figure and is expressed in square units.

→ Volume can be calculated for 3d figures like cube, cuboid, cone, etc….

→ Area of some known figures.
a) Triangle = \(\frac{1}{2}\)bh
b) Square = side × side
c) Rectangle = length × breadth
d) Equilateral triangle = \(\frac{\sqrt{3}}{4}\)a²
e) Trapezium = \(\frac{\mathrm{h}}{2}\) (a + b)
where a, b are lengths of parallel sides
f) Circle = πr²
g) Parallelogram = bh

→ Perimeter of
a) Triangle = sum of all sides
b) Square = 4 × side = 4s or 4a
c) Rectangle = 2(l + b)
d) Equilateral triangle = 3s
e) Circle = 2πr²
f) Parallelogram = sum of lengths of all four sides.

Students can go through AP 8th Class Maths Notes Chapter 10 Visualising Solid Shapes to understand and remember the concepts easily.

Class 8 Maths Chapter 10 Notes Visualising Solid Shapes

→ Recognising 2D and 3D objects.

→ Recognising different shapes in nested objects.

→ 3D objects have different views from different positions.

→ A map is different from a picture.

→ A map depicts the location of a particular object/place in relation to other objects/ places.

→ Symbols are used to depict the different objects/places.

→ There is no reference or perspective in a map.

→ Maps involve a scale which is fixed for a particular map.

→ For any polyhedron, F + V- E = 2
where ‘F’ stands for number of faces, V stands for number of vertices and E stands for number of edges. This relationship is called Euler’s formula.

→ Plane Shape : A flat figure or a closed two dimensional figure like triangle, square, rectangle, polygon, elipse, circles are called plane shapes..In other words-they also
called 2D – Shapes.

These are basic plane shapes.

→ Solid Shapes: The shapes which have 3 dimensions that are length, breadth, height/ depth are called solid shapes. They are also called 3d shapes.
Examples :

Students can go through AP 8th Class Maths Notes Chapter 9 Algebraic Expressions and Identities to understand and remember the concepts easily.

Class 8 Maths Chapter 9 Notes Algebraic Expressions and Identities

→ Expressions are formed from variables and constants.

→ Terms are added to form expressions. Terms themselves are formed as product of factors.

→ Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively. In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative integers as exponents) is called a polynomial.

→ Like terms are formed from the same variables and the powers of these variables are the same, too! Coefficients of like terms need not be the same.

→ While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.

→ There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.

→ A monomial multiplied by a monomial always gives a monomial.

→ While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the monomial.

→ In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial). Note that in such multiplication, we may get terms in the product which are like and have to be combined.

→ An identity is an equality, which is true for all values of the variables in the equality. On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.

→ The following are the standard identities:
(a + b)² = a² + 2ab + b² (I)
(a – b)² = a² – 2ab + b² (II)
(a + b) (a – b) = a² – b² (III)

→ Another useful identity is (x + a) (x + b) = x² + (a + b) x + ab (IV)

→ The above four identities are useful in carrying out squares and products of algebraic expressions. They also allow easy alternative methods to calculate products of numbers and so on.

→ Variable : A symbol and place holder for a quantity that may change (or) A symbol, usually a letter standing in for an unknown numerical value is called a variable.

→ x, y, z …….. are examples for variables
Usually for real number unknowns x, y are used for time (t) radius (r) distance (s or d) etc., are used in general.

→ Constant : A digit which has fixed values are called constants.
Example 4, 5, – 10, – 100, etc.
Unknown constants are a, b, c, ……..

→ Term : A constant or variable or product of any of them is called term, (but no variable having negative powers, decimal powers)
for example : 4, 4x, 4xy, x²yz, 8.5 yz², -10x, 10 – x, etc., are terms.
Note : (variables having negative integers as exponents are not terms)
but \(\frac{4}{x}\), \(\frac{x^2}{y}\), z^-3 etc., are not counted as
terms because x, y (variables are in denominaters and z^-3 has negative power(-3).
So not considered as terms.

→ Expression : An expression is a combination (set) of terms combined using operations +, -, x.
Ex : 2x – y ; 4x², \(\frac{2}{3}\)x⁵, x⁴ + y³ – 3z;
4p² + 3pq + 5q² etc., are examples for expressions.

→ Factors and coefficient:
In the term xyz its factors are x, y, z
In the term 4x its factors are 4, x
In the term – y its factors are -1, y
Thus 7, x, y are factors of the term (7xy) numerical factors are called “numerical coefficient” or “coefficient”

→ Monomials: An expression having only one term is called Monomial.
Ex:.4x, -5x², 7x²y³z, 10y, -9m

→ Binomial : An expression having only two terms is called Binomial.
Ex : 4x – y (4x, – y are two terms)
5x² – 3yz² (5x², – 3yz² are two terms) 15xyz² + 3x (15xyz², 3x)
but 4x – y² is not a Binomial because there is only one term (4x) ; y^-2 is not a term.

→ Trinomial : An expression having 3 terms is called trinomial.
Ex: x² + y² – z² ; here (x², y², – z² are 3 terms)
4xy – y³ – z⁴ ; x – 6y + 7z, xyz + y + z² … etc

→ Polynomial: An expression having one or more than one terms is called a poly-nomial.
Ex: p + q + 2r + 3s ;
4x²y²z² ; 2x + 3y – 17z …. etc.
→ Expressing an expression on number line :

→ (2x) on number line

→ 2x – 3

‘p’ shows 2x – 3

→ x + 5

‘p’ indicates (x + 5)

→ 4x + 4

Students can go through AP 8th Class Maths Notes Chapter 8 Comparing Quantities to understand and remember the concepts easily.

Class 8 Maths Chapter 8 Notes Comparing Quantities

→ Discount is a reduction given on marked price.
Discount = Marked Price – Sale Price.

→ Discount can be calculated when discount percentage is given.
Discount = Discount % of Marked Price

→ Additional expenses made after buying an article are included in the cost price and are known as overhead expenses.
CP = Buying price + Overhead expenses

→ Sales tax is changed on the sale of an item.by the government and is added to the Bill Amount. Sales tax = Tax% of Bill Amount

→ GST stands for Goods and Services Tax and is levied on supply of goods or services or both.

→ Compound interest is the interest calculated on the previous year’s amount (A = P + I)

→ (i) Amount when interest is compounded annually
= P(1 + \(\frac{\mathrm{R}}{100}\))ⁿ ; P is principal, R is rate of interest, n is time period
(ii) Amount when interest is compounded half yearly
P(1 + \(\frac{\mathrm{R}}{200}\))²ⁿ{ \(\frac{R}{2}\) is half yearly rate and 2n = number of ‘half-years’

→ We use different methods to compare quantities.

→ Ratio and percentage are two such methods used to compare things / quantities.

→ In a ratio we can compare any number of quantities in their order (having same units)
For example : In a cricket team
3 all rounders, 2 bowlers, 4 and the 5 are batsman and 1 wicket keepr are there then ratio of bowlers, all rounders, batsman and wicket keepers
= 2 : 3 : 5 : 1

→ In a school bag 5 textbooks, 4 notes, 3 workbooks are kept.
then ratio of workbooks, school notes, and textbooks are 3 : 4 : 5

→ In a class of 36 students 20 are boys then ratio of Girls and boys are …………….

ratio of girls and boys =16 : 20
If the ratio has only two things, then we can write it in fraction from = \(\frac{16}{20}\) = \(\frac{4}{5}\) (by simplication)
Thus a: b can be written as (\(\frac{\mathrm{a}}{\mathrm{~b}}\)) also.

→ We can express this ratio form in percentage form also.

→ Percentage means, considering it to the quantity out of ‘100’.

→ For example : A student scored 16 out of 20 then the same can be calculated for out of 100 (using unitary method)
Here Out of 20 score is 16
then out of 100 (= 20 × 5)
score will be 16 × 5 = 80
So it means he iscores 80 out of 100 which can be expressed as 16 : 20 (or) \(\frac{16}{20}\) (or) 80%

→ Expressing in percentages is a good habbit for comparison.

→ To convert a fraction into percentage we have to multiply that fraction with 100.
Example : 1) \(\frac{4}{5}\) then \(\frac{4}{5}\) × 100 = 80%
So \(\frac{4}{5}\) means 80%
2) \(\frac{1}{4}\) then \(\frac{1}{4}\) × 100 = 25%
So 1 : 4 or \(\frac{1}{4}\) or 25%

→ To convert given percentage into fraction we divide it with 100 and simplify it.
Example : 40% = \(\frac{40}{100}\) = \(\frac{2}{5}\) So 40% = \(\frac{2}{5}\).
Convert 35% into fraction.
35% = \(\frac{35}{100}\) = \(\frac{7}{20}\) ; So 35% = \(\frac{7}{20}\).
Now checking \(\frac{7}{20}\) into percentage to convert the fraction into percentage, we have to mulitply it with 100.
So \(\frac{7}{20}\) × 100 = 7 × 5 = 30% hence verified.

Students can go through AP 8th Class Maths Notes Chapter 7 Cubes and Cube Roots to understand and remember the concepts easily.

Class 8 Maths Chapter 7 Notes Cubes and Cube Roots

→ Numbers like 1729, 4104, 13832, are known as Hardy- Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.

→ Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1,8,27,… etc.

→ If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube.

→ The symbol \(\sqrt[3]{ }\) denotes cube root. For example \(\sqrt[3]{27}\) = 3.

→ If we are able to express a number as a sum of two cubes of two positive integers then such number can be called as Hardy – Ramanujan
For example :
1. 4104 is Hardy – Ramanujan number because 4104 can be expressed as sum of 2³ and 16³
Check :
4104 = 2³ + 16³ [∵ 2³ = 8 ; 16³ = 16 × 16 × 16 = 256 × 16 = 4096]
= 8 + 4096 = 4104
So 4104 is sum of two cubes ; so it is Hardy – Ramanujan (or)
4104 = 9³ + 15³ [9³ = 9 × 9 × 9 = 729 15³ = 15 × 15 × 15 = 3375]
= 729 + 3375
= 4104
So 4104 can be expressed as either 2³ + 16³ (or) 9³ + 15³

Ex. 2. 13832 = 18³ + 20³ [18³ = 18 × 18 × 18 = 5832 = 20³ = 20 × 20 × 20 = 8000
= 5832 + 8000
= 13832
(or)
13832 = 2³ + 24³ [2³ = 2 × 2 × 2 = 8 24³ = 24 × 24 × 24 = 13824]
= 8 + 13824
= 13832
So we can call 13832 is a Hardy – Ramanujan number.
G.H. Hardy is British Mathematician Ramanujan is Indian Mathematician. Another example is 20683, which can be expressed as 27³ + 10³ (or) 24³ + 19³
Check :
20683 = 27³ + 10³ [∵ 27³ = 19683 ; 10³ = 1000]
= 19683 + 1000
= 20683
(or)
20683 = 24³ + 19³ [24³ = 13824 ; 19³ = 6859]
= 13824 + 6859
= 20683

Clarification : If 9 a Hardy – Ramanujan number
9³ = 1³ + 2³ but we cannot express 9 in another way of sum of two cubic numbers. Hence ‘9’ is not a Hardy – Ramanujan number.
H – Ramanujan numbers are also called as Taxi – cab number
Numbers that are the smelliest num-ber that can be expressed as sum of two cubes in ‘n’ distinict ways are described as Taxi – cab numbers.
Example :
1729 = 1³ + 12³ [1³ = 1, 12³ = 1728]
(or) = 9³ + 10³ [9³ = 729, 10³ = 1000]
Solid figures : Figures which have 3 – dimensions are known as solid figures.

1 unit solid ; bolded sides in the above figure are its 3 – dimensions.

Solid showing with 2 units on each side total 8 units = 2 × 2 × 2 = 2³
So this is a 2 unit cube have 8 cubic units

This a 3 units cube which have 3 × 3 × 3 = 27 units.
Perfect cubes or Cube numbers } A number which can be expressed as product of 3 same numbers.
Ex : 1, 8, 27, 64, 125 ………….. are cube numbers.
Why because
1 can be expressed as 1 × 1 × 1
8 can be expressed as 2 × 2 × 2
27 can be expressed as 3 × 3 × 3
64 can be expressed as 4 × 4 × 4
125 can be expressed as 5 × 5 × 5
First 4 cube numbers below 100 are 1, 8, 27, 64
First 9 cube numbers below 1000 are 1, 8, 27, 64, 125, 216, 343, 512, 729
and it is clear that we have 10 perfect cubes upto 1000.
Filling the table showing cubes of natural numbers 1 to 10.

Natural number(n)	1	2	3	4	5	6	7	8	9	10
Corresponding Cubic number n3	1	8	27	64	125	216	343	512	729	1000

Natural number(n) 1 2 3 4 5 6 7 8 9 10
Corresponding Cubic number n3 1 8 27 64 125 216 343 512 729 1000
From the above table we can conclude that cube of an even number is also even
Ex : (2 – 8) (4 – 64) (6 – 216) (8 – 512) …………
and cube of an odd number is also odd
Ex : (3 – 27) (5 -125) (7 – 343) (9 – 729)

Students can go through AP 8th Class Maths Notes Chapter 6 Squares and Square Roots to understand and remember the concepts easily.

Class 8 Maths Chapter 6 Notes Squares and Square Roots

→ If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number.

→ All square numbers end with 0, 1, 4, 5, 6 or 9 at units place.

→ Square numbers can only have even number of zeros at the end.

→ Square root is the inverse operation of square.

→ There are two integral square roots of a perfect square number.
Positive square root of a number is denoted by the symbol √ .
For example, 3² = 9 gives √9 = 3

→ Square number: Any natural number multiplied by itself gives a square number as a result. Consider ‘6’ a natural number.
When multiplied by itself we get 6 × 6 = 36, then 36 is a square number (or) perfect square.
Consider another natural number 13
then multiply it with itself = 13 × 13 = 169
then 169 is a square number (or) perfect square.
Thus 1, 4, 9, 16, 25 ……………. are examples for perfect squares.

Thus 1, 4, 9,16, 25, 36, 49, 64, 81 are 9 perfect squares below 100.

Students can go through AP 8th Class Maths Notes Chapter 5 Data Handling to understand and remember the concepts easily.

Class 8 Maths Chapter 5 Notes Data Handling

→ Data mostly available to us in an unorganised form is called raw data.

→ In order to draw meaningful inferences from any data, we need to organise the data systematically.

→ Frequency gives the number of times that a particular entry occurs.

→ Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.

→ Grouped data can be presented using histogram. Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.

→ Data can also presented using circle graph or pie chart. A circle graph shows the relationship between a whole and its part.

→ There are certain experiments whose outcomes have an equal chance of occurring.

→ A random experiment is one whose outcome cannot be predicted exactly in advance.

→ Outcomes of an experiment are equally likely if each has the same chance of occurring.

→ Probability of an event = \(\frac{\text { Number of outcomes that make an event }}{\text { Total number of outcomes of the experiment }}\), when the outcomes are equally likely

→ One or more outcomes of an experiment make an event.

→ Chances and probability are related to real life.

→ Particular information of a group is called data.
For example :

1. No. of people above age of 60 years of a village is a data.
2. Total production of wheat/paddy in a district is (Mandal/Village Wise) a data.
3. Rainfall recordings related to a climate zone is another data.
4. Wickets taken by a bowler against a team in IPL/World Cup season wise is also data.

Like this any particular information related to a group is called ‘Data’.
To understand more/to plan further/ to expect the future and for many instances, the study of this DATA.is more and more useful.
Now a days and nearest future we are going to deal with ‘data banks’ like our currency banks.
The study of this data is called ‘Data Science’.
The person who is expert in this data science is called ’Data Scientist’.
One who analyse this data to interpret more is called ’Data analyser’.
These are very common words we are looking now-a-days.
When a data is represented by symbols then it is called as ‘pictograph’ (pictograph).
Many times it is easy and good to see data in graphically.
A Pictograph gives us a quick under-standing and we can quick respond to many questions related to a pictograph.

→ Bar graph or Histogram: A Bargraph is a display of information using bars of uniform width and their heights being proportional to the respective values. Bar heights give the quantity for each category and Bars are of equal width with equal gaps in between.

→ Double Bar graph : A bar graph show¬ing two sets of data simultaneuously is called double bar graph. It is for quick comparison.
For example :
No. of girls and boys class wise in a year No. of overs bowled Vs No. of extra runs gives.
No. of marks in each subject in SAI and SAII
These types of information can be dis-played in double bargraphs.