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**Quantitative Aptitude**Offline Downloads:
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UNIT 3
SQUARE-SQUARE
ROOT AND
CUBE-CUBE ROOT
(A)
Main Concepts and Results
• A natural number is called a perfect square if it is the square of
some natural number.
i.e., if m = n2, then m is a perfect square where m and n are natural
numbers.
• A natural number is called a perfect cube if it is the cube of some
natural number.
i.e., if m = n3, then m is a perfect cube where m and n are natural
numbers.
• Number obtained when a number is multiplied by itself is called
the square of the number.
• Number obtained when a number is multiplied by itself three times
are called cube number.
• Squares and cubes of even numbers are even.
• Squares and cubes of odd numbers are odd.
• A perfect square can always be expressed as the product of pairs of
prime factors.
• A perfect cube can always be expressed as the product of triplets
of prime factors.

MATHEMATICS
• The unit digit of a perfect square can be only 0, 1, 4, 5, 6 or 9.
• The square of a number having:
1 or 9 at the units place ends in1.
2 or 8 at the units place ends in 4.
3 or 7 at the units place ends in 9.
4 or 6 at the units place ends in 6.
5 at the units place ends in 5.
• There are 2n natural numbers between the squares of numbers n
and n+1.
• A number ending in odd numbers of zeroes is not a perfect square.
• The sum of first n odd natural numbers is given by n2.
• Three natural numbers a, b, c are said to form a pythagorean
triplet if a2 + b2 = c2.
• For every natural number m > 1, 2m, m2–1 and m2 + 1 form a
pythagorean triplet.
• The square root of a number x is the number whose square is x.
Positive square root of a number x is denoted by
x.
• The cube root of a number x is the number whose cube is x. It is
denoted by
3
x.
• Square root and cube root are the inverse operations of squares
and cubes respectively.
• If a perfect square is of n digits, then its square root will have
n
2
⎛ n +1⎞
digit if n is even or ⎜
⎟ digit if n is odd.
⎝ 2 ⎠
• Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9
end with digits 0, 1, 4, 5, 6 and 9 respectively.
Think
74
and
Discuss
1.
Describe what is meant by a perfect square. Give an example.
2.
Explain how many square roots a positive number can have. How are
these square roots different?
EXEMPLAR PROBLEMS

UNIT-3
Key
Concept
To be Noted
SQUARE ROOTS
Words
A square root of a number n is a number m which, when
multiplied by itself, equals n.
Numbers
The square roots of 16 are 4 and – 4 because 42 = 16 and (– 4)2 =
16.
Algebra
If m2 = n, then m is a square root of n.
Think
and
Discuss
1.
Which type of number has an exact square root?
2.
Which type of number has an approximate square root?
3.
How can we use perfect squares to estimate a square root, such as 8 ?
• Cube of the number ending in 2 ends in 8 and cube root of the
number ending in 8 ends in 2.
• Cube of the number ending in 3 ends in 7 and cube root of the
number ending in 7 ends in 3.
(B) Solved
Examples
In examples 1 to 7, out of given four choices only one is correct. Write
the correct answer.
Example 1 : Which of the following is the square of an odd number?
(a) 256
Solution
(b) 361
(c) 144
(d) 400
: Correct answer is (b).
Example 2 : Which of the following will have 1 at its units place?
(a) 192
Solution
(b) 172
(c) 182
(d) 162
: Correct answer is (a).
Example 3 : How many natural numbers lie between 182 and 192?
(a) 30
Solution
(b) 37
(c) 35
(d) 36
: Correct answer is (d).
SQUARE-SQUARE ROOT AND CUBE-CUBE ROOT
75

MATHEMATICS
Example 4 : Which of the following is not a perfect square?
(a) 361
Solution
(b) 1156
(c) 1128
(d) 1681
: Correct answer is (c).
Example 5 : A perfect square can never have the following digit at
ones place.
(a) 1
Solution
(b) 6
(c) 5
(d) 3
: Correct answer is (d).
Example 6 : The value of 176 + 2401 is
(a) 14
Solution
(b) 15
(c) 16
: Correct answer is (b).
( 176 +
2401 = 176 + 49 = 225 = 15
Example 7 : Given that 5625 =75, the value of
(a) 82.5
Solution
(d) 17
(b) 0.75
)
0.5625 + 56.25 is:
(c) 8.25
(d) 75.05
: Correct answer is (c).
If ( 5625 = 75, then
0.5625 = 0.75 and
56.25 = 7.5)
In examples 8 to 14, fill in the blanks to make the statements true.
Example 8 : There are __________ perfect squares between 1 and 50.
Solution
: 6
Example 9 : The cube of 100 will have __________ zeroes.
Solution
: 6
Example 10 : The square of 6.1 is ____________.
Solution
76
: 37.21
1.
Squaring a number and taking a square root are inverse operations.
What other inverse operations do you know?
2.
When the factors of a perfect square are written in order from the least
to greatest, what do you notice?
3.
Why do you think numbers such as 4, 9, 16, ... are called perfect
squares?
4.
Suppose you list the factors of a perfect square. Why is one factor
square root and not the other factors?
EXEMPLAR PROBLEMS

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