Worksheet Solutions: A Square and A Cube | Mathematics (Maths) Class 8 PDF Download

Multiple Choice Questions

125²=125×125
So, 3√125²=  3√5×5×5×5×5×5

=25

Q2: Which of the following is not a perfect square number?

(a) 1156
(b) 4657
(c) 4624
(d) 7056

Ans: B)

A perfect square is a number that can be expressed as the square of an integer.

Check if each option is a perfect square:

• $1156=34\mathrm{<sup\; style="font-size:\; 12px\; !important">2</sup>}1156\; =\; 34^2$
• $46574657$ is not a perfect square (not an integer square root).
• $4624=68\mathrm{<sup\; style="font-size:\; 12px\; !important">2</sup>}4624\; =\; 68^2$
• $7056=84\mathrm{<sup\; style="font-size:\; 12px\; !important">2</sup>}7056\; =\; 84^2$

Q3: Which of these is not a perfect cube but a perfect square of a number 
(a) 729
(b) 2197
(c) -1331
(d) 169
Ans: (d)

169= 13×13

Q4: Which of these is a perfect cube
(a) 216
(b) 392
(c) 8640
(d) 243
Ans: (a)

Q5: √0.09 is
(a) 0.3
(b) 0.03
(c) 0.9
(d) 0.33
Ans: A)

We need to find the square root of 0.09 by the long division method.

Square Root of 9 is 3.

Further, we can do 3/10=0.3

Thus square root of 0.09 is 0.3.

Q6: The sum of first n odd natural numbers is
(a) n²
(b) 2n
(c) n²+1
(d) n²−1
Ans: A)

The sum of the first n odd natural numbers is known to be n².

Q7: The one’s digit of the cube of 53 is:

(a) 9
(b) 3
(c) 7
(d)  1

Ans: (c)

Q8: The volume of the cube is 5832m³ , the side is
(a) 18m
(b) 16 m
(c) 28 m
(d) 19m

^{Ans: (a)}

Q9:A perfect square can never have the following digit in its ones place
(a) 8
(b) 4
(c) 0
(d) 1
Ans: A)

Sol: The last digit of a perfect square can be 0, 1, 4, 5, 6, or 9.
8 is not included in this list.

Q10: Which of the following expressions is not a perfect cube
(a) 27 x 125 x 64
(b) 1331 x 125 x 8
(c) 15 x 8 x 25 x 9
(d) None of these
Ans: (d)

A perfect cube is a number that can be expressed as the cube of an integer. Since all integers in options a, b,and  c can be expressed as a cube of integers. Therefore,  None of these is the answer.

True and False

Q2: 999 is a perfect cube.
Ans: False

Q3: The square of a number is positive, so the cube of that number will also be positive.
Ans: False

Q4: 125×8×27 is a perfect cube.
Ans: True

Q5: For an integer p, p³ is always greater than p²_.
Ans: False

Q6: 8³ = 5.12
Ans: False

Fill in the blanks

Q1: The square of an even number is _____
Ans: even
Sol: The square of an even number is always even.

Q2: The least number by which 72 be multiplied to make it a perfect cube is __________.
Ans: 3

Q3: √4096 is ____
Ans: 64

Q3: If 8x³=216, then x ix ____
Ans: 3

Q3: The cube of .5 is _____
Ans: .125

Q4: There are _________ perfect cubes between 1 and 1000.
Ans: 8

Q5: The digit at the ones place of 23³ is ____
Ans: 7

Q6: The cube of the even natural number is ____
Ans: even

Find the cube roots by prime factorization method

Q1: 15625
Ans: 15625=5×5×5×5×5×5

³√15625=25

Q2: 2744
Ans: 2744=2³×7^3
3√2744=2×7=14

Q3: 125/2197
Ans: 125/2197=(5x5x5)/(13×13×13)
³√125/2197=5/13

Q4: 5832
Ans: 5832= 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3
³√5832=2×3×3=18

Q5: 64000
Ans: 64000 = 4 x 4 x 4 x 10 x 10 x 10
³√64000=4×10=40

Find the square roots by prime factorization method

Q1: 144
Ans: 144 = 2 × 2 × 2 × 2 × 3 × 3
Pairing: (2×2) (2×2) (3×3)
√144 = 2 × 2 × 3 = 12

Q2: 3600
Ans: 3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5
Pairing: (2×2) (2×2) (3×3) (5×5)
√3600 = 2 × 2 × 3 × 5 = 60

Q3: 2025
Ans: 2025 = 3 × 3 × 3 × 3 × 5 × 5
Pairing: (3×3) (3×3) (5×5)
√2025 = 3 × 3 × 5 = 45

Q4: 81/256
Ans: 81/256 = (3 × 3 × 3 × 3) / (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2)
Pairing numerator: (3×3) (3×3) → gives 3 × 3
Pairing denominator: four pairs of (2×2) → gives 2 × 2 × 2 × 2
√(81/256) = (3 × 3) / (2 × 2 × 2 × 2) = 9/16

Q5: 1024
Ans: 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (ten 2s)
Pairing: five pairs of (2×2)
√1024 = 2 × 2 × 2 × 2 × 2 = 32

Q6: 3844
Ans: 3844 = 2 × 2 × 31 × 31
Pairing: (2×2) (31×31)
√3844 = 2 × 31 = 62

Answer the following questions

Q1: Is 256 a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.
Ans: 256=2⁸=2³×2³×2²
Therefore, it is not a perfect cube
So, it must be divided by 4 to become a perfect cube

Q2: What could be the possible ‘one’s’ digits of the square root of each of the following numbers?
(i) 1801
(ii) 856
(iii) 1008001
(iv) 6577525
Ans: (i) Last digit is 1 , So one digit can be 1 or 9 as 1²=1 and 9²=81
(ii) Last digit is 5 , So one digit can be 4 or 6 as 4²=164 and 6²=36
(iii) Last digit is 1 , So one digit can be 1 or 9 as 1²=1 =1 and 9²=81
(iv) Last digit is 5 , So one digit will be 5 as 5²=25

Q3: Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. Find the numbers.
Ans: Let the number be x,2x,3x
Now
x³+8x³+27x³=4500
36x³=4500

x³=4500/36
x³=125
x=5
So the numbers are 5,10,15

Q4: Find the smallest number by which following number must be divided to get a perfect square. Also, find the square root of the perfect square so obtained.
(i)600
(ii)2904
Ans: (i) 600= 2 x 2 x 2 x 3 x 5 x 5
We can see numbers 2 and 3 is not in pairs, So to make a perfect square, it has to be divided by 6
Then number will become=100
Now
100= 2 x 2 x 5 x 5
√100=2×5=10
(ii)2904=2 x 2 x 2 x 3 x 11 x 11
We can see number 2 and 3 is not in pair, So to make a perfect square, it has to be divided by 6
Then number will become=484
Now
484= 2 x 2 x 11 x 11
√484=2×11=22

Q5: Find the smallest number by which following number must be multiplied to get a perfect square. Also, find the square root of the perfect square so obtained.
(i) 1008
(ii) 1280
(iii) 1875
Ans:
(i)1008= 2 x 2 x 2 x 2 x 3 x 3 x 7
We can see number 7 is not in pair, So to make a perfect square, it has to be multiplied by 7
Then number will become=7056
Now
7056= 2 x 2 x 2 x 2 x 3 x 3 x 7 x 7
√7056=2×2×3×7=84

(ii)1280=2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5
We can see number 5 is not in pair, So to make a perfect square, it has to be multiplied by 5
Then number will become=6400
Now
6400= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5
√6400=2×2×2×2×5=80

(iii)1875= 3 x 5 x 5 x 5 x 5
We can see number 3 is not in pair, So to make a perfect square, it has to be multiplied by 3
Then number will become=5625
Now
5625= 3 x 3 x 5 x 5 x 5 x 5
√5625=3×5×5=75