8th Class

 Squares and Square Roots  

• Square: The square of a number is the product obtained when a number is multiplied by itself.

 

• Perfect Square: A perfect squares are the shares of whole numbers. Perfect squares are formed by multiplying a whole number by itself.

 

• Properties of Squares:

(i) A number ending in 2, 3, 7 or 8 is never a perfect square. All square numbers end in 0, 1,4,5,6 or 9. (ii) A number ending in an odd number of zeroes is never a perfect square. (iii) Square numbers have only even number of zeros at the end. (iv) Squares of even numbers are even. (v) Squares of odd numbers are odd. (vi) For every natural number \[n,{{\left( n+1 \right)}^{2}}\text{ }-{{n}^{2}}=\left( n+1 more...

 Cubes and Cube Roots  

• Cubes

(i) The cube of a number is the product of the number multiplied by itself twice. (ii) Write the cube of a number using the cube symbol or notation. (iii) \[{{8}^{3}}\]is read as 'eight cubed' or 'the cube of eight', or 'eight to the power of three.  

• Estimating the cubes of numbers

Estimate the cube of a number by determining the range in which its value lies. e.g. Estimate the cube of 10.6 by determining the range in which its value lies.   Solution 10 < 10.6 < 11 \[\leftarrow \] Determine the range \[103<{{\left( 10.6 \right)}^{3}}<113\]\[\leftarrow \]Cube the range \[1000<{{\left( 10.6 \right)}^{3}}<1331\]Estimated answer \[\therefore {{\left( 10.6 \right)}^{3}}\]is between 1000 and 1331.  

• Perfect cube

(i) A more...

 Comparing Quantities  

• Compound interest: Amount at compound interest is given by \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\], where,

A - Amount,   P - Principal,    R - Rate of interest, n - Time period. (i) Compound interest = A - P (ii) In case of depreciation (or) decay, \[A=P{{\left( 1-\frac{R}{100} \right)}^{n}}\]

• If the rates of increase in population P are p%, q% and r% during 1st, 2nd and 3rd years respectively, then the population after 3 years =

\[=P\left( \frac{P}{100} \right)\left( 1+\frac{q}{100} \right)\left( 1+\frac{r}{100} \right)\].  

• If principal = R.s P, rate = R% per annum and time = n years, then

(a) Amount after 'n' years (compounded annually) is \[A=P{{\left( 1+\frac{R}{100} \right)}^{n}}\]   (b) Amount after 'n' years (compounded half-yearly) is \[A=P{{\left( 1+\frac{R}{2\times 100} \right)}^{2n}}\] where more...

 Algebraic Expressions and Identities  

• A combination of constants and variables connected by +, -, x and - is known as an algebraic expression.

e.g.,  \[2-3x+5{{x}^{-2}}{{y}^{-1}}+\frac{x}{3{{y}^{3}}}\]  

• Polynomial:

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial. e.g., \[2-3x+5{{x}^{2}}{{y}^{-1}}-\frac{x}{3}x{{y}^{3}}\]  

• Like terms: Terms formed from the same variables whose powers are same are called like

terms. The coefficients of like terms need not be the same.  

• Unlike terms: Terms formed from different variables whose powers may be same or different are called unlike terms. The coefficients of unlike terms may or may not be the same.

In other words, terms with the same variables and which have the same exponent are more...

 Visualising Solid Shapes  

• Geometrical shapes:

Plane shapes have two measurements - length and breadth and therefore they are called two-dimensional shapes. e.g.,   Solid objects have three measurements – length, breadth and height or depth. So, they are called three – dimensional shapes. Also, Solids Occupy some Space. e.g.,     ·                

• Description more...

 Mensuration  

• Perimeter: The length of the boundary of a plane figure is called its perimeter.

 

• Area: The amount of surface enclosed by a plane figure is called its area.

 

• Rectangle: Given a rectangle of length T units and breadth 'b' units,

  (i) Perimeter of the rectangle \[=2\left( l+b \right)\] units (ii) Diagonal of the rectangle,\[~d=\sqrt{{{l}^{2}}+{{b}^{2}}}\] units (iii) Area of the rectangle \[=\text{ }\left( \text{l}\times b \right)\]sq. units (iv)\[\operatorname{Length}=\left( \frac{area}{breadth} \right)units\] (v) \[\operatorname{Breadth}=\left( \frac{area}{length} \right)units\]  

• Area of four walls of room: Let there be a room with length T units, breadth 'b' units and height 'h' units.

Then (i) Area of four walls \[=2\left( l+2 \right)\times h\]sq. units more...

 Exponents and Powers  

• Exponential equation: An equation which has an unknown quantity as an exponent is called an exponential equation.

e.g.,      (i)\[{{5}^{x}}=625\] (ii) \[{{3}^{x-5}}=1\]   Note: If ax = ay, than x = y.  

• Standard form of numbers: A number written in the form \[\left( m\times {{10}^{n}} \right)\]is said to be in standard form if 'm' is a decimal number between 1 and 9 and 'n' is either a positive or a negative integer.

Very large numbers and very small numbers are expressed in standard form.  

• Laws of exponents (Integers): For any two non-zero integers 'a' and 'b', and any integers 'm' and 'n', the following laws hold good.

(i) \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]                  (ii)\[\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m+n}}\left( m>n \right)\]             (iii) \[\frac{{{a}^{m}}}{{{a}^{n}}}=\frac{1}{{{a}^{m+n}}}\left( m<n more...

 Direct and Inverse Proportions  

• Unitary method:

A method in which the value of a quantity is first obtained to find the value of any required quantity is called unitary method.  

• Direct proportion:

(i) 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. (ii) 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 \[{{\operatorname{y}}_{1}}\,and\,{{y}_{2}}\] are the values of y corresponding to the values \[{{\operatorname{x}}_{1\,}}\,and\,{{x}_{2}}\]of z respectively then \[\frac{{{x}_{1}}}{{{y}_{1}}}=\frac{{{x}_{2}}}{{{y}_{2}}},\]

 

• Examples:

(a) As the number of articles more...

 Factorisation  

• Factorisation:

(i) The process of writing an algebraic expression as the product of two or more algebraic expressions is called factorisation. (ii) When we factories an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions. (iii) An irreducible factor is that which cannot be expressed further as a product of factors. (iv) A systematic way of factorising an expression is the common factor method. It consists of three steps: (a) Write each term of the expression as a product of irreducible factors. (b) Look for and separate the common factors and (c) Combine the remaining factors in each term in accordance with the distributive law. (v) Sometimes, all the terms in a more...

  Introduction to Graphs  

• Bar graph: A bar graph is used to show comparison among categories.

 

• Pie graph: A pie graph is used to compare parts of whole.

 

• Histogram: Representation that shows data in intervals.

 

• Line graph: It shows data that changes continuously over periods of time.

 

• Linear graph: A straight line graph is called a linear graph.

 

• The Cartesian system:

(i)A plane is divided into 4 quarters (called quadrants) by two perpendicular lines, intersecting at 0 (called origin). The horizontal line is called the X-axis and the vertical line is called the Y-axis. (ii) A point is represented by the horizontal distance from the origin called the x-coordinate and more...