Topics
Rational Numbers
- Rational Numbers
- Closure Property of Rational Numbers
- Commutative Property of Rational Numbers
- Associative Property of Rational Numbers
- Distributive Property of Multiplication Over Addition for Rational Numbers
- Identity of Addition and Multiplication of Rational Numbers
- Negative Or Additive Inverse of Rational Numbers
- Concept of Reciprocals or Multiplicative Inverses
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
- Multiples and Common Multiples
Linear Equations in One Variable
- Constants and Variables in Mathematics
- Equation in Mathematics
- Expressions with Variables
- Word Problems on Linear Equations
- Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Solving Equations Having the Variable on Both Sides
- Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
- Reducing Equations to Simpler Form
- Equations Reducible to Linear Equations
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Basic Concept of Polygons
- Classification of Polygons
- Properties of Quadrilateral
- Sum of Interior Angles of a Polygon
- Sum of Exterior Angles of a Polygon
- Quadrilaterals
- Properties of Trapezium
- Properties of Kite
- Properties of a Parallelogram
- Properties of Rhombus
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The adjacent angles in a parallelogram are supplementary.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Properties of Rectangle
- Properties of a Square
- Property: The diagonals of a square are perpendicular bisectors of each other.
Data Handling
Practical Geometry
- Geometric Tool
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Some Special Cases
Squares and Square Roots
- Concept of Square Number
- Properties of Square Numbers
- Some More Interesting Patterns of Square Number
- Finding the Square of a Number
- Concept of Square Roots
- Finding Square Root Through Repeated Subtraction
- Finding Square Root Through Prime Factorisation
- Finding Square Root by Division Method
- Square Root of Decimal Numbers
- Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
- Ratio
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Calculation of Interest
- Concept of Compound Interest
- Deducing a Formula for Compound Interest
- Rate Compounded Annually Or Half Yearly (Semi Annually)
- Applications of Compound Interest Formula
Algebraic Expressions and Identities
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Classification of Terms in Algebra
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Exponents and Powers
Visualizing Solid Shapes
Direct and Inverse Proportions
Factorization
- Factors and Common Factors
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Concept of Find the Error
Introduction to Graphs
Playing with Numbers
Definition
Exponents: An exponent is a numerical notation that indicates the number of times a number is to be multiplied by itself. Exponent is also called as power or index. Example, 75.
Notes
Introduction of Exponents and powers:
Distance between Sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is 1,439,000,000,000 m.
We can’t read such numbers. These very large numbers are difficult to read, understand, and compare. To make these numbers easy to read, understand, and compare, we use exponents.
1. Exponents:
-
An exponent is a numerical notation that indicates the number of times a number is to be multiplied by itself.
-
Exponent is also called as power or index.
-
The main purpose of learning exponents is to write a very big number or a larger number in the simplest form.
-
10, 000 = 10 × 10 × 10 × 10 = 104.
The short notation 104 stands for the product 10 × 10 × 10 × 10.
104 is called the exponential form of 10,000.
-
Generally, exponent says that how many times the base should be multiplied.
If 75 is given the meaning is 7 is multiplied 5 times i.e. 7 × 7 × 7 × 7 × 7.
-
Here ‘7’ is called the base and ‘5’ the exponent.
-
The number 75 is read as 7 raised to the power of 5 or simply as the fifth power of 7.
- Generally, if a is any number, a × a × a ×.......... (m times) = am
Read am as ‘a raised to the power m’ or ‘the mth power of a’.
Here m is a natural number.
∴ 54 = 5 × 5 × 5 × 5 = 625. Or, the value of the number 54 = 625.
Base with negative integer:
-
(– 10)3 = (– 10) × (– 10) × (– 10) = 100 × (– 10) = – 1000
So, (–1)odd number = - 1 -
(–5)4 = (– 5) × (– 5) × (– 5) × (– 5) = 25 × 25 = 625
So, (–1)even number = + 1
2. Comparison of quantities using exponents:
-
If two numbers in standard form have the same power of 10, then the number with the larger factor is greater.
For example, 8.72 × 1024 < 9.4326 × 1024.
-
If two numbers in standard form have the same factor, then the number with the larger power of 10 will be greater.
For example, 8.72 × 108 > 8.72 × 104. -
If two numbers in standard form have different factors and the different power of 10, then the number with the higher power of ten (i.e., the larger exponent) is the larger number.
For example, 8.72 × 1024 > 9.4326 × 1023.
Example
Which one is greater 23 or 32?
We have, 23 = 2 × 2 × 2 = 8 and 32 = 3 × 3 = 9.
Since 9 > 8,
so, 32 is greater than 23.
Example
Expand a3b2, a2b3, b2a3, b3a2. Are they all same?
a3b2 = a3 × b2 = (a × a × a) × (b × b) = a × a × a × b × b
a2b3 = a2 × b3 = a × a × b × b × b
b2a3 = b2 × a3 = b × b × a × a × a
b3a2 = b3 × a2 = b × b × b × a × a
a3b2 and a2b3 the powers of a and b are different. Thus a3b2 and a2b3 are different.
a3b2 and b2a3 are the same, since the powers of a and b in thesetwo terms are the same. The order of factors does not matter.
Thus, a3b2 = a3 × b2 = b2 × a3 = b2a3. Similarly, a2b3 and b3a2 are the same.
Example
Express the following number as a product of powers of prime factor: 432
432
= 2 × 216 = 2 × 2 × 108
= 2 × 2 × 2 × 54
= 2 × 2 × 2 × 2 × 27
= 2 × 2 × 2 × 2 × 3 × 9
= 2 × 2 × 2 × 2 × 3 × 3 × 3
or 432 = 24 × 33.
Example
Express the following number as a product of powers of prime factor: 16000
16,000
= 16 × 1000
= (2 × 2 × 2 × 2) × 1000
= 24 × 103 .......(as 16 = 2 × 2 × 2 × 2)
= (2 × 2 × 2 × 2) × (2 × 2 × 2 × 5 × 5 × 5) = 24 × 23 × 53.......(Since, 1000 = 2 × 2 × 2 × 5 × 5 × 5)
= (2 × 2 × 2 × 2 × 2 × 2 × 2 ) × (5 × 5 × 5)
or, 16,000 = 27 × 53.
Example
Work out (1)5, (–1)3, (–1)4, (–10)3, (–5)4
i) (1)5 = 1 × 1 × 1 × 1 × 1 = 1
ii) (–1)3 = (– 1) × (– 1) × (– 1) = 1 × (– 1) = – 1
iii) (–1)4 = (–1) × (–1) × (–1) × (–1) = 1 ×1 = 1
iv) (–10)3 = (–10) × (–10) × (–10) = 100 × (– 10) = – 1000
v) (–5)4 = (–5) × (–5) × (–5) × (–5) = 25 × 25 = 625
