Quantitative Aptitude · Mathematics

Squares and Square Roots

196 Questions

Squares and square roots questions assess numerical ability through perfect squares, digit properties, and fast calculation techniques. Many competitive exams feature Vedic mathematics shortcuts to solve these quickly. Mastering these rules improves overall calculation speed.

Perfect square propertiesVedic math squaresFinding square rootsDigit replacement puzzlesLeast perfect squares

Squares and Square Roots Questions

Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

Find the squares of the following numbers without actual multiplication:
$49$
$52$

  1. $2441; 2704$
  2. $2401; 2784$
  3. $2401; 2704$
  4. $2441; 2784$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$49^2=(40+9)^2=40(40+9)+9(40+9)$
$\;\;\;\;\;=(40)^2+40\times9+9\times40+9^2$
$\;\;\;\;\;=1600+360+360+81$
So, $49^2=2401$
$52^2=(50+2)^2=50(50+2)+2(50+2)$
$\;\;\;\;\;\;=50^2+50\times2+2\times50+2^2$
$\;\;\;\;\;\;=2500+100+100+4$
So, $52^2=2704$.

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Which of the following is not a perfect square?

  1. $16384$
  2. $23857$
  3. $18496$
  4. $11025$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$16384= 128\times 128$         (perfect square)

$18496=136\times 136$           (perfect square)
$11025=105\times 105$         (perfect square)
$23857 = 1\times  23857$        (prime no. so not a perfect square)
Hence, option B is correct.

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the atleast number which must  be added to each of the following numbers to get a perfect square. Also find the square root of the perfect square numbers.


$a)525$

  1. 4

  2. 3

  3. 1

  4. 6

  5. 12
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$i)\ 23^2=529$
$\therefore \ $ it will add $4$ to $525$ we get $529$
which is payout square
$\therefore \ 525+4=529=23^2$

$\therefore 4$ should be added


Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square Also find the square root of the perfect square so obtained 
$(i) 402, (ii) 1989, (iii) 3250, (iv) 825, (v) 4000$

  1. <span class="block ng-binding">Least number which must be subtracted:$(i) 2, (ii) 22, (iii) 1, (iv) 21, (v) 52$

    Square root of the perfect square:
    $(i) 20, (ii) 34 ,(iii) 55, (iv) 26, (v) 67$
  2. <span class="block ng-binding">Least number which must be subtracted:$(i) 2, (ii) 53, (iii) 1, (iv) 41, (v) 31$

    Square root of the perfect square:
    $(i) 20, (ii) 44, (iii) 57, (iv) 28, (v) 63$
  3. <span class="block ng-binding">Least number which must be subtracted:$(i) 6, (ii) 22, (iii) 50, (iv) 31, (v) 40$

    Square root of the perfect square:
    $(i) 19, (ii) 41, (iii) 49 ,(iv) 27 ,(v) 65$
  4. <span class="block ng-binding">Least number which must be subtracted:$(i) 8, (ii) 41, (iii) 12, (iv) 56, (v) 4$

    Square root of the perfect square:
    $(i) 19 ,(ii) 22, (iii) 37, (iv) 26, (v) 61$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For $402$

Nearest perfect square is $400$
$\sqrt{400}$ $=$ $20$
$402$ $-$ $400$ $=$ $2$
For $1989$
Nearest perfect square is $1936$

$\sqrt{1936}$ $=$ $44$
$1936$ $-$ $1989$ $=$ $53$

For $3250$
Nearest perfect square is $3249$

$\sqrt{3249}$ $=$ $57$
$3250$ $-$ $3249$ $=$ $1$
For $825$
Nearest perfect square is $784$
$\sqrt{784}$ $=$ $28$
$825$ $-$ $784$ $=$ $41$
For $4000$
Nearest perfect square is $3969$
$\sqrt{3969}$ $=$ $63$
$4000$ $-$ $3969$ $=$ $31$
From this, Option B is correct answer.

Multiple choice maths squares and square roots approximation of square roots estimating square roots square root of non perfect squares

Mr. Hansraj wants to find the least number of boxes to be added to get a perfect square. He already has $7924$ boxes with him. How many more boxes are required?

  1. $819$
  2. $412$
  3. $419$
  4. $176$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Find square root by long division method.

$\therefore \sqrt {7924}$ = $89.01$

Hence, the perfect square number smaller than $7924$ is
$89^2 = 7921$
The next perfect square no. is $90^2 = 8100$

So, $8100 - 7924 = 176$
Therefore, the man needs $176$ more boxes in order to get a perfect square number.
So, option D is correct.