Page 283 - ICSE Math 8

SQUARES AND SQUARE ROOTS

2
Its square will be x × x = x .
Also known as perfect square Methods of Finding Square Roots
If x is any number: 1. By Repeated Subtraction method
Its square root will be x . 2. By Prime Factorization method
2
If x = m, then m = x. 3. By Long Division method

Test for Perfect Squares: Prime
Factorization Method Square Root of a Fraction = Square root of the numerator
Find the prime factors of the given Square root of the denominator
number and then pair them up. If all
the factors are paired, then the number Approximation of Square Roots by Long Division Method
is a perfect square. 1. If n decimal places are required, then there should be 2n
digits in the decimal part.
2. The square root is calculated upto (n + 1) decimal places
and then rounded off.

Properties of Perfect Squares
1. A number with 2, 3, 7 or 8 at its units place is never a perfect square.
2. A number ending in an odd number of zeros is never a perfect square.
3. The square of an even number is always even and that of an odd number is always odd.
4. The square of a proper fraction is always smaller than the fraction.
5. The difference in squares of any two consecutive natural numbers is equal to their sum.
6. The square of a natural number n is equal to the sum of the first n-odd natural numbers.

CUBES AND CUBE ROOTS

2
Its cube will be x × x × x = x . The cube of a negative integer is always negative.
Cube of the numerator
Also known as perfect cube The cube of rational number =
Cube of the denominator
If x is any number:
Methods of Finding Cube Roots
3
Its cube root will be x . 1. By Prime Factorization method
3
3
If x = m, then m = x. 2. By using units digits
Test for Perfect Cubes: Prime Factorization 1. Cube Root of Negative Perfect Cube is
Method negative of the cube root of its absolute value.
Find the prime factors of the given number and 2. xy = 3 x × 3 y
3
then group them in triplets. If all the factors are
grouped, then the number is a perfect cube. p 3 p
3. 3 =
q 3 q
Properties of Perfect Cubes
Cubes of the numbers ending with:
• 1,4,5,6 or 9 also end with the same digits.
• 2, end with 8 and vice versa.
• 3, end with 7 and vice versa.
• 0, end with three zeros.

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