PPT: Squares-Square Roots-Cubes & Cube Roots | Quantitative Techniques for CLAT PDF Download

 Page 1


SQUARES, SQUARE 
ROOTS, CUBES, CUBE 
ROOTS
Page 2


SQUARES, SQUARE 
ROOTS, CUBES, CUBE 
ROOTS
Squares and Perfect 
Squares
What Are Squares?
When the exponent 
of a base is 2, the 
numbers obtained 
are called squares or 
squared numbers.
Perfect Squares
A natural number is 
called a perfect 
square if it is the 
square of a number. 
Examples include 1, 
4, 9, 16, 25, and 36, 
which are perfect 
squares of 1, 2, 3, 4, 
5, and 6 respectively.
Non-Perfect Squares
Numbers like 5, 6, 7, 8, 11, 50, and 125 are not 
perfect squares of any natural number.
Page 3


SQUARES, SQUARE 
ROOTS, CUBES, CUBE 
ROOTS
Squares and Perfect 
Squares
What Are Squares?
When the exponent 
of a base is 2, the 
numbers obtained 
are called squares or 
squared numbers.
Perfect Squares
A natural number is 
called a perfect 
square if it is the 
square of a number. 
Examples include 1, 
4, 9, 16, 25, and 36, 
which are perfect 
squares of 1, 2, 3, 4, 
5, and 6 respectively.
Non-Perfect Squares
Numbers like 5, 6, 7, 8, 11, 50, and 125 are not 
perfect squares of any natural number.
Determining a Perfect Square
Step I: Prime 
Factorization
Express the given 
number as the 
product of its 
prime numbers.
Step II: Pair 
Formation
Express the 
prime factors as 
pairs of the same 
number.
Step III: Check 
for Unpaired 
Factors
Examine if there 
are any unpaired 
factors 
remaining.
Step IV: 
Conclusion
If there is no 
unpaired factor, 
the given number 
is a perfect 
square.
Page 4


SQUARES, SQUARE 
ROOTS, CUBES, CUBE 
ROOTS
Squares and Perfect 
Squares
What Are Squares?
When the exponent 
of a base is 2, the 
numbers obtained 
are called squares or 
squared numbers.
Perfect Squares
A natural number is 
called a perfect 
square if it is the 
square of a number. 
Examples include 1, 
4, 9, 16, 25, and 36, 
which are perfect 
squares of 1, 2, 3, 4, 
5, and 6 respectively.
Non-Perfect Squares
Numbers like 5, 6, 7, 8, 11, 50, and 125 are not 
perfect squares of any natural number.
Determining a Perfect Square
Step I: Prime 
Factorization
Express the given 
number as the 
product of its 
prime numbers.
Step II: Pair 
Formation
Express the 
prime factors as 
pairs of the same 
number.
Step III: Check 
for Unpaired 
Factors
Examine if there 
are any unpaired 
factors 
remaining.
Step IV: 
Conclusion
If there is no 
unpaired factor, 
the given number 
is a perfect 
square.
Properties of Squares
Even and Odd Numbers
Squares of even numbers are even, while squares of 
odd numbers are odd.
Terminal Digits
A number ending with 2, 3, 7, or 8 is never a perfect 
square. Numbers ending with an odd number of zeros 
cannot be perfect squares either.
Divisibility Properties
The square of a natural number (other than 1) is either 
a multiple of 3 or exceeds a multiple of 3 by 1. 
Similarly, it's either a multiple of 4 or exceeds a 
multiple of 4 by 1.
Consecutive Squares
The difference between the squares of two 
consecutive natural numbers is equal to their sum. For 
example, 5² - 4² = 25 - 16 = 9, which equals 5 + 4 = 9.
Page 5


SQUARES, SQUARE 
ROOTS, CUBES, CUBE 
ROOTS
Squares and Perfect 
Squares
What Are Squares?
When the exponent 
of a base is 2, the 
numbers obtained 
are called squares or 
squared numbers.
Perfect Squares
A natural number is 
called a perfect 
square if it is the 
square of a number. 
Examples include 1, 
4, 9, 16, 25, and 36, 
which are perfect 
squares of 1, 2, 3, 4, 
5, and 6 respectively.
Non-Perfect Squares
Numbers like 5, 6, 7, 8, 11, 50, and 125 are not 
perfect squares of any natural number.
Determining a Perfect Square
Step I: Prime 
Factorization
Express the given 
number as the 
product of its 
prime numbers.
Step II: Pair 
Formation
Express the 
prime factors as 
pairs of the same 
number.
Step III: Check 
for Unpaired 
Factors
Examine if there 
are any unpaired 
factors 
remaining.
Step IV: 
Conclusion
If there is no 
unpaired factor, 
the given number 
is a perfect 
square.
Properties of Squares
Even and Odd Numbers
Squares of even numbers are even, while squares of 
odd numbers are odd.
Terminal Digits
A number ending with 2, 3, 7, or 8 is never a perfect 
square. Numbers ending with an odd number of zeros 
cannot be perfect squares either.
Divisibility Properties
The square of a natural number (other than 1) is either 
a multiple of 3 or exceeds a multiple of 3 by 1. 
Similarly, it's either a multiple of 4 or exceeds a 
multiple of 4 by 1.
Consecutive Squares
The difference between the squares of two 
consecutive natural numbers is equal to their sum. For 
example, 5² - 4² = 25 - 16 = 9, which equals 5 + 4 = 9.
Interesting Facts About Squares
Pythagorean Triplets
For every natural number m 
(other than 1), there is a 
triplet (2m, m²-1, m²+1) 
called a Pythagorean Triplet. 
These triplets obey the 
relation (2m)² + (m²-1)² = 
(m²+1)². For example, when 
m = 2, we get the familiar 4, 
3, and 5, where 4² + 3² = 5².
Unit Digits
The square of any natural 
number will have either 0, 1, 
4, 5, 6, or 9 at the unit place. 
This can be useful for 
quickly checking if a number 
might be a perfect square.
Sum of Odd Numbers
The square of a natural 
number is always equal to 
the sum of the first n odd 
natural numbers. For 
example, 4² = 1 + 3 + 5 + 7 = 
16, and 6² = 1 + 3 + 5 + 7 + 9 
+ 11 = 36.