Complete the Ganita Prakash Class 8 Worksheet and NCERT Class 8 Maths Chapter 1 A Square and A Cube Worksheet with Answers before your unit tests for better preparation.
A Square and A Cube Worksheet Class 8
Class 8 Maths A Square and A Cube Worksheet
A Square and A Cube Class 8 Ganita Prakash Worksheet
Princess Mrinalini inherited a beautiful palace garden with 150 lamps arranged In a straight line. On the night of the grand festival of Diwali, she announced a challenge to all the festival guests. In the challenge:
- Each guest is assigned a number from 1 to 150.
- Guest 1 walks through the garden and switches on every lamp.
- Guest 2 comes next and switches off every 2nd lamp.
- Guest 3 toggles every 3rd lamp. (turns it off if it’s on, turns it on if it’s off)
- Guest 4 toggles every 4th lamp.
- This continues until Guest 150 toggles only the last lamp.
Question 1.
Answer the following questions.
(a) How many different guests will touch lamp number 36 as they walk through the garden?
Answer:
9 guests
(b) After all the guests are done, will lamp number 50 be glowing or not? Explain why.
Answer:
No
(c) Princess Mrinalini noticed lamp number 64 stays glowing at the end. What makes this lamp different from lamp number 60?
Answer:
64 is a perfect Square
(d) Out of the first 20 lamps, which ones will still glow after the festival guests have all finished?
Answer:
Lamps Numbers:1, 4,9 and 16
(e) Out of 150 lamps in the garden, which lamp numbers will be glowing at the end of the process? How many lamps glow in total?
Answer:
Lamps number: 4, 4, 9, 16, 25, 36, 49, 64, 81 ,100, 121, 144 total 12 lamps glow.
Square Numbers
Question 2.
Here is an incomplete table of squares of numbers from 11 to 25. Based on the above keys complete it.
| 112 = 11 × 11 = 121 | 182= _____ × _____ = 324 |
| 122 = _____ × _____ = _____ | _____ = _____ × _____ = 361 |
| 132 = _____ × _____ = _____ | _____ = 20 × 20 = _____ |
| _____ = 14 × 14 = 196 | 212 = 21 × 21 = _____ |
| _____ = _____ × _____ = 225 | 222 = _____ × _____ = _____ |
| 162 = 16 × 16 = _____ | 232 = _____ × _____ = 529 |
| 172 = 17 × 17 = 289 | _____ = 24 × 24 = 576 |
Answer:
| 112 = 11 × 11 = 121 | 182 = 18 × 18 = 324 |
| 122 = 12 × 12 = 144 | 192 = 19 × 19 = 361 |
| 132 = 13 × 13 = 169 | 202 = 20 × 20 = 400 |
| 142 = 14 × 14 = 196 | 212 = 21 × 21 = 441 |
| 152 = 15 × 15 = 225 | 222 = 22 × 22 = 484 |
| 162= 16 × 16 = 256 | 232 = 23 × 23 = 529 |
| 172 = 17 × 17 = 289 | 242 = 24 × 24 = 576 |
| 252 = 25 × 25 = 625 |
Question 3.
Arav’s mother baked cupcakes and arranged them neatly in a square box. Each row had 12 cupcakes, and the box had the same number of rows and columns. How many cupcakes can fit in one box?
Answer:
144
Patterns and properties of perject squares
Question 4.
For each case, decide whether the number described ‘can’ or ‘cannot’ be a perfect square. (Write Yes or No)
(a) I am a 3-digit number and my unit digit is 7.
Answer:
No
(b) I am a 4-digit number and my unit digit is 9.
Answer:
Yes
(c) I am a 2-digit number and my unit digit is 8.
Answer:
No
(d) I am a 5-digit number and my unit digit is 2.
Answer:
No
(e) I am a 4-digit number and my unit digit is 6.
Answer:
Yes
(f) I am a 4-digit number and my unit digit is 5.
Answer:
yes
Question 5.
What will be the ones digit of the square of the following numbers?
(a) 42
(b) 53
(c) 28
(d) 79
Answer:
(a) 4
(b) 9
(c) 4
(d) 1
Question 6.
Which of the following pairs of numbers have the same ones digit as in their square numbers?
(a) 16 and 24
(b) 35 and 50
(c) 24 and 38
(d) 19 and 23
Answer:
(a) 16 and 24
Question 7.
Which one among 342, 582, 922, and 562 has last digit 4?
Answer:
582 and 922
Question 8.
Find the difference between 1052and 952.
Answer:
2000
Question 9.
The sum of two consecutive square numbers is 313. Find the numbers.
Answer:
144, 169
Perfect Squares and Odd Numbers
Question 10.
Show that the sum of the first 7 odd numbers is 72.
Answer:
Do it yourself
Question 11.
Write the sum of the first 5 odd numbers and verify the sum.
Answer:
52 = 25
Question 12.
Find the sum of first:
(a) 17 odd numbers.
(b) 21 odd numbers.
Answer:
(a) 289
(b) 441
Question 13.
Find the sum of the following without adding them,
(a) 1 + 3 + 5 + 7 + 9 + 11
(b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
(c) 11 + 13 + 15 + 17 + … + 29 + 31
Answer:
(a) 36
(b) 36
(c) 121
(d) 231
Question 14.
Show by successive subtraction of odd numbers that 81 is a perfect square.
Answer:
Do it yourself
Question 15.
Check whether the following numbers are perfect square numbers or not. (Write Yes or No)
(a) 148
(b) 321
Answer:
(a) No
(b) No
Perfect Squares and Triangular Numbers
Question 16.
Write down all perfect square numbers from 1 to 100 and represent them as sum of two triangular numbers.
Answer:
1 = 1 + 0
4 = 1 + 3
9 = 3 + 6
16 = 1 + 15
25 = 10 + 15
36 = 15 + 21
49 = 21 + 28
64 = 28 + 36
81 = 36 + 45
100 = 45 + 55
Square Roots
Radhika wanted to design a square-shaped flower garden in her backgard. She had flowerpots, tiles, and measuring tapes ready with her. Her elder brother Aarav decided to help her.
Let us see how they are working for their garden.
Question 17.
Radhika placed 49 flowerpots in a square arrangement. How many flower pots are there in each row?
Answer:
7
Question 18.
Aarav sald, area of this gardert LS 144 m2 and we need to fence it. What should be the Length of one side of the garden?
Answer:
12 m
Question 19.
Radhika decided to decorate one corner of her square-shaped garden with tiles. The total area of that corner is 6,400 cm2. If side of each square tile is of 2 cm, how many tiles will she need to completely cover that corner?
Answer:
1600
Question 20.
Find the square roots of the following numbers by successive subtraction of odd numbers.
(a) 49
(b) 64
(c) 100
(d) 121
Answer:
(a) 7
(b) 8
(c) 10
(d) 11
Question 21.
Find the square roots of the following numbers using prime factorisation method,
(a) 196
(b) 324
(c) 576
(d) 1225
(e) 3136
(f) 5184
Answer:
(a) 14
(b) 18
(c) 24
(d) 35
(e) 56
(f) 72
Question 22.
Estimate the value of the following square roots using the nearest perfect squares.
(a) √50
(b) \(\sqrt{2025}\)
Answer:
(a) 7
(b) 45
Question 23.
Which of the following numbers are perfect squares? Also, find their square roots.
(a) 484
(b) 529
Answer:
(a) 484,22
(b) 529,23
(c) 1089,33
Question 24.
Find the smallest square number, that Is divisible by each of the numbers: 5, 6 and 7.
Answer:
44100
Question 25.
A square park has an area of 6084 m2. What is the length of its each side? If a fence is to be built around it, how many metres of fencing wire is required?
Answer:
78m, 312m
Question 26.
Find the smallest number by which 2408 must be multiplied so that the product is a perfect square. Also find the square root of the product.
Answer:
602,104
Question 27.
Find the smallest number by which 62208 must be divided so that the quotient obtained is a perfect square. Also, find the square root of the quotient.
Answer:
3,144
Question 28.
A square garden has an area between 9600 m2 and 9800 m2. Estimate the possible side length of-the garden without calculating the actual square root.
Answer:
Between 98 m and 99 m
Cubic Numbers
There was a kingdom known as the Cubcia. Unlike other kingdoms, this land was guarded by magical gates that only opened with the power of perfect cubes. One day, a curious boy named Kabir set out to explore the kingdom.
At the entrance, a glowing voice announced: “Only those who understand the magic of cubes may enter.” The first gate showed the number 8. The voice asked: “What is the smallest cube greater than 1?”
Kabir quickly replied: 13 = 1, 23 = ______.
The cube of 2 is ______.
The gate opened with a loud creak!
Inside, he found the Garden of Numbers, where trees bore fruits shaped like cubes.
A wise owl perched on a tree challenged him “To move ahead, tell me which numbers here are perfect cube numbers.” The owl pointed to fruits labelled: 27, 30, 64, 125, 150, and 216.
Question 29.
Help Kabir to Identify the perfect cube numbers among the given numbers.
Answer:
27,64 and 216
Question 30.
Write the cubes of the following numbers:
(a) 17
(b) 23
(c) 100
Answer:
(a) 4713
(b) 12167
(c) 1000000
(d) \(\frac{8}{27}\)
Question 31.
The annual fair preparations were in full swing. Radhika, Aarav, Meera, and Akmad were busy in the storeroom, arranging cube-skaped boxes filled with chocolates, toys, and stationeries.
(a) Meera found a carton shaped like a cube with side length 15 cm. What is its volume?
Answer:
3375 m2
(b) The teacher pointed at a cube-shaped box with side length 20 cm. Its volume ends with which digits?
Answer:
Zero
(c) Aarav arranged a stack of cube boxes in the shape of a longer cube with 6 boxes along each edge. If each small cube has a side length of 2 cm. How many small cubes are there in total? What is the volume of the larger cube?
Answer:
216, 1728 cm2
Question 32.
Is 36 a cube? Check.
Answer:
No
Question 33.
List all possible unit digits of cubes. Which digits never appear at the end of a cube?
Answer:
0,1, 2, 3, 4, 5, 6, 7, 8, 9, (No digit is absent)
Question 34.
If the cube of a number ends in 7, what must be the last digit of the number itself?
Answer:
3
Question 35.
Write the unit digit of 433. Verify by finding its actual cube.
Answer:
7
Perfect Cubes and Consecutive Odd Numbers
Question 36.
Write the sum of following consecutive odd numbers without doing the addition. Use properties of cubes.
(a) 43 + 45 + 47 + 49 + 51 + 53 + 55 =
(b) 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109 =
(c) 133 + 135 + 137 + 139 + 141 + 143 + 145 + 147 + 149 + 151 + 153 + 155 =
Answer:
(a) 343
(b) 1000
(c) 1728
Cube Roots
Question 37.
Check whether the following numbers are perfect cubes or not. (Write Yes or No)
(a) 512
(b) 1024
(c) 3375
Answer:
(a) Yes
(b) No
(c) Yes
Question 38.
Find the cube roots of the following numbers using prime factorisation.
(a) 1728
(b) 5832
(c) 9261
Answer:
(a) Yes
(b) No
(c) Yes
Question 39.
A huge storage cubical box contains 13,824 of unit cubes. Find the sidelength of the cubical box.
Answer:
24 unit
Worksheet On A Square and A Cube Class 8
A. Choose the correct option.
1. Which of the following is not a perfect square?
(a) 324
(b) 529
(c) 289
(d) 159
Answer:
(d) 159
2. The cube of 15 is:
(a) 3755
(b) 3750
(c) 3375
(d) 3315
Answer:
(c) 3375
3. Which of the following cannot be the unit’s digit of a perfect square?
(a) 6
(b) 9
(c) 3
(d) 5
Answer:
(c) 3
4. Which one of the following numbers is a perfect cube?
(a) 392
(b) 729
(c) 343
(d) Both (b) and (c)
Answer:
(d) Both (b) and (c)
5. The smallest number by which 540 must be multiplied to make it a perfect square is:
(a) 5
(b) 6
(c) 15
(d) 2
Answer:
(a) 5
6. Which of the following pairs are perfect cubes?
(a) (125, 216)
(b) (343, 512)
(c) (100, 225)
(d) (27, 50)
Answer:
(a) (125, 216)
7. The smallest 3-digit perfect square number is:
(a) 100
(b) 121
(c) 144
(d) 169
Answer:
(a) 100
8. Which of these numbers has an integer square root as well as an integer cube root?
(a) 8
(b) 64
(c) 121
(d) 144
Answer:
(b) 64
Directions. (For Q.9 -10): In the following questions, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option as:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.
9. Assertion (A): 576 is a perfect square.
Reason (R): A number is a perfect square if its all prime factors occur in even powers.
Answer:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
10. Assertion (A): 512 is not a perfect cube.
Reason (R): A number is a perfect cube if its all prime factors occur in multiples of three.
Answer:
(d) Assertion (A) is false but Reason (R) is true.
Fill in the blanks.
1. The square of 125 is ______, and its square root is ______.
Answer:
15625, 11.18
2. The cube root of 3375 is ______.
Answer:
15
3. The difference between consecutive perfect squares is always an ______ number.
Answer:
Odd
4. The square root of 1225 is ______.
Answer:
35
5. The cube of 20 is ______.
Answer:
8000
C. State whether the following statements are true (T) or false (F).
1. Every perfect square has exactly one square root.
Answer:
Flase
2. A perfect cube can end with exactly two zeros.
Answer:
Flase
3. The cube of any odd number is always odd.
Answer:
True
4. 1156 is a perfect square.
Answer:
True
5. The product of two perfect squares is always a perfect square.
Answer:
True
D. Solve the following.
Question 1.
The area of a square lawn is 324 m2. Find the length of its side.
Answer:
18 m
Question 2.
A cube-shaped container has a volume of 9261 cm3 . Find its edge length.
Answer:
21 m
Question 3.
A cloth of area 98O1 cm2 is cut into square handkerchLefs of maximum size. Find the side Length of each handkerchief. ALso, find the number of handkerchiefs.
Answer:
99 cm, 1(One)
Question 4.
Ria wants to make a cube using small dice of side length 1 cm. If she uses total 512 dice, what is the side length of the big cube formed?
Answer:
8 cm
Question 5.
In Delhi a school is preparing for a Math’s expo. For this, they choose a square field of area 6400 m2. At the centre of the field, a cube-shaped hall is built with a sidelength of 12 m. For decoration, the organisers stack 343 small cubes (each of side 1 cm) to form a big cube. Based on the above information, answer the following questions:
(a) What is the sidelength of the square field?
Answer:
80 m
(b) What is the volume of cube-shaped hall?
Answer:
1728 m3
(c) What is the sidelength of the big cube used for decoration?
Answer:
7 cm