Sqares & Square roots Shyam Prasad Sahoo Class VIII sec e Roll No 31 School No 5363 DAV Cspur ,BBSR

Properties of Square Numbers We have learnt about various types of numbers such as Natural Numbers, Whole Numbers, Integers and Rational Numbers.

Properties of Square Numbers Definition :Numbers that can be expressed as the square of a number are called square numbers or perfect squares. If a and b are two natural numbers such that a = b2,  then a is called a square number or a perfect square of b. Every natural number has a square, but every natural number is not a perfect square itself. Numbers that cannot be expressed as the square of another number are not perfect squares. Since there are infinite natural numbers, there are infinite numbers of perfect squares.

Properties of Square Numbers All perfect squares: Have 0, 1, 4, 5, 6 or 9 in their units place. Never have 2, 3, 7 or 8 in their units place. The numbers that have 0, 1, 4, 5, 6 or 9 in their units place maybe perfect squares whereas the numbers that have 2, 3, 7 or 8 in their units place are never perfect squares. 

Properties of Square Numbers All numbers ending with 1 or 9 have 1 in the units place of their squares. All squares having 1 in their units place are squares of the numbers ending with either 1 or 9. All numbers ending with 2 or 8 have 4 in the units place of their squares

Properties of Square Numbers All squares having 4 in their units place are squares of the numbers ending with either 2 or 8. All numbers ending with 3 or 7 have 9 in the units place of their squares. All squares having 9 in their units place are squares of the numbers ending with either 3 or 7.

Properties of Square Numbers All numbers ending with 4 or 6 have 6 in the units place of their squares. All squares having 6 in their units place are squares of the numbers ending with either 4 or 6. All numbers ending with 5 have 5 in the units place of their squares. All squares having 5 in their units place are squares of the numbers ending with 5. All numbers ending with 0 have 0 in the units place of their squares. All squares having 0 in their units place are squares of the numbers ending with 0.

Properties of Square Numbers Square of a number ending with zero(s) contains double the number of zeroes than the number. All square numbers contain an even number of zeroes. Odd square numbers are squares of numbers ending with 1, 3, 5, 7 or 9. Even square numbers are squares of numbers ending with 0, 2, 4, 6 or 8.

Properties of Square Numbers The numbers whose dot patterns can be arranged as triangles are called triangular numbers. The sum of any two consecutive triangular numbers is a square number.

Properties of Square Numbers For two consecutive numbers n and n+1, there are 2n non square numbers between n2 and (n+1)2. Sum of first n odd natural numbers is n2. The square of any odd number can be written as the sum of two consecutive positive integers.

Properties of Square Numbers

Finding Square and Square roots We know that a number m is called a square number if it is expressed as n2. Here n is called the square root of m. Square root is the inverse operation of squares. Every square number is a sum of the first n odd natural numbers.

Finding Square roots Square root of a given number is a number whose square is equal to the given number. Positive square root of a number is denoted by the symbol √.  Square root of a number can be found using the following three methods: Repeated Subtraction Method Prime Factorisation Method Long Division Method

Finding Squares

Pythagorean Triplet

Finding Square roots

Finding Square roots

Finding Square and Square roots

Cubes and Cube Roots Shyam Prasad Sahoo Class VIII sec E Roll No 31, School no 5363 DAV Cspur BBSR

Cubes and Cube Roots If a and b are two natural numbers such that a3 = b,  then b is called the cube of a. If the units digit of a3 is b, then the cubes of all numbers ending with a will have their units digit as b.

Properties of Cubes and Cube Roots The cubes of all numbers that end in 2 have 8 as the units digit. The cubes of all numbers that end in 3 have 7as the units digit.

Properties of Cubes and Cube Roots Definition: The cube root of a given number is a number, which, when multiplied with itself three times, gives the number. The first odd natural number is the cube of 1. The sum of the next two odd natural numbers is the cube of 2. The sum of the next three odd natural numbers is the cube of 3, and so on. If a given number is a perfect cube, then its prime factors will always occur in groups of three. The cube root of a number can be found using the prime factorisation method or estimation method.

Properties of Cubes and Cube Roots

Properties of Cubes and Cube Roots

Properties of Cube Roots