1. The odd thing about the numbers 13 and 31 is that their squares, 169 and 961, are mirror reflections of one another. Can you locate two more similar pairs?
1. The odd thing about the numbers 13 and 31 is that their squares, 169 and 961, are mirror reflections of one another. Can you locate two more similar pairs?
Explanation:
The first couple are 12 and 21.
Numbers 122 = 144 and 212 = 441 are squares.
The next set is 102 and 201.
Numbers 1022 = 10404 and 2012 = 40401 are squares.
2. Put three distinct numbers in the circles so that you will always get a perfect square when you add the numbers at the conclusion of each line.
2. Put three distinct numbers in the circles so that you will always get a perfect square when you add the numbers at the conclusion of each line.
Explanation:
The three numbers 6, 19, and 30 are the only ones where adding the ends of each line results in a perfect square.
6 + 19 = 25
6 + 30 = 36
19 + 30 = 49
3. There are four digis in a perfect square number, and none of them are zero. The values of the digits are even, even, odd, and even from left to right. Look up the number.
3. There are four digis in a perfect square number, and none of them are zero. The values of the digits are even, even, odd, and even from left to right. Look up the number.
Explanation:
assuming PQRS is a perfect square
In this scenario, P, Q, R, and S are all even.
8836 is the ideal square, then.
4. {(62 + (82)1/2)}3
4. {(62 + (82)1/2)}3
Explanation:
Given, {(62 + (82)1/2)}3
= {(36 + (64)1/2)}3
= {(36 + (8)}3
= {44}3
= 44 × 44 × 44
= 85,184
5. {(52 + (122)1/2)}3
5. {(52 + (122)1/2)}3
Explanation:
Given, {(52 + (122)1/2)}3
= {(25 + (144)1/2)}3
= {(25 + (12)}3
= {37}3
= 37 × 37 × 37
= 50,653
6. Consider the following: 327 + 30.008 + 30.064
6. Consider the following: 327 + 30.008 + 30.064
Explanation:
= 3√27 + 3√0.008 + 3√0.064
= 3√(3 × 3 × 3) + (3√(8/1000)) + 3√(64/1000)
= 3√(33) + (3√(2 × 2 × 2)/(10 × 10 × 10)) + 3√((4 × 4 × 4)/(10 × 10 × 10))
= 3 + (3√(23/103) + 3√(43/103)
= 3 + (2/10) + (4/10)
= 3 + 0.2 + 0.4
= 3.6
7. By using the process of repeated subtraction, determine the square root of 324.
7. By using the process of repeated subtraction, determine the square root of 324.
Explanation:
The question makes it clear that 324
Now, beginning with 1, we deduct each subsequent odd number as follows:
324 – 1 = 323
323 – 3 = 320
320 – 5 = 315
315 – 7 = 308
308 – 9 = 299
299 – 11 = 288
288 – 13 = 275
275 – 15 = 260
260 – 17 = 243
243 – 19 = 224
224 – 21 = 203
203 – 23 = 180
180 – 25 = 155
155 – 27 = 128
128 – 29 = 99
99 – 31 = 68
68 – 33 = 35
35 – 35 = 0
Here, 324 is reduced to 0 by taking away 18 odd numbers.
Consequently, 18 is the square root of 324.
8. A square plot has a 37m2 area
Calculate the width of the plot's one side.
8. A square plot has a 37m2 area
Calculate the width of the plot's one side.
Explanation:
The question makes it clear that,
Area of a square plot: 38m2 = 40401/400 from NCERT Exemplar Class 8 Maths Solutions Chapter 3
As we already know, side2 = area of square plot.
(40401/400) = side2
Square roots are calculated on both sides.
Side = √(40401/400)
= √((201 × 201)/(20 × 20))
= √(2012/202)
= 201/20
9. Find the smallest square number that each of the numbers 8, 9, and 10 can divide by.
9. Find the smallest square number that each of the numbers 8, 9, and 10 can divide by.
Explanation:
First, we must determine the LCM for 8, 9, and 10.
LCM of 8, 9, and 10 is therefore 2 2 2 3 3 5.
= 360
Now, when the components are combined, (2 2) × 2 × (3 × 3) × 5
Specifically, 2 and 5 are unable to form a pair.
As a result, 360 must be multiplied by 2 5 = 10 to make it perfectly square.
Then,
= 360 × 10 = 3600
3600 is the least square number that can be divided perfectly into 8, 9, and 10.
10. The sum of the squares of three numbers in the ratio 2:3:5 is 608.
10. The sum of the squares of three numbers in the ratio 2:3:5 is 608.
Explanation:
Assume that the three numbers are 2a, 3a, and 5a.
Then,
The square root of three numbers, as stated, is 608
i.e. (2a)2 + (3a)2 + (5a)2 = 608
4a2 + 9a2 + 25a2 = 608
38a2 = 608
a2 = 608/38
a2 = 16
a = √16
a = √(4 × 4)
a = √(42)
a = 4
∴These are the figures: 2a = 2 4 = 8.
3a = 3 × 4 = 12
5a = 5× 4 = 20
11. A monarch wanted to honour his advisor, a learned person in the realm. In order to name his own reward, he asked the wise guy. The wiseman thanked the monarch but declared that his only request for the next month would be for a few gold coins every day. For a total of 30 days, the coins were to be distributed as follows: one coin on the first day, three coins on the second day, five coins on the third day, etc. Find out how much coins the advisor will receive throughout that month without using a calculator.
11. A monarch wanted to honour his advisor, a learned person in the realm. In order to name his own reward, he asked the wise guy. The wiseman thanked the monarch but declared that his only request for the next month would be for a few gold coins every day. For a total of 30 days, the coins were to be distributed as follows: one coin on the first day, three coins on the second day, five coins on the third day, etc. Find out how much coins the advisor will receive throughout that month without using a calculator.
Explanation:
It is clear from the question that,
The advisor receives 1 + 3 + 5 + coins overall.
The sequence of odd natural numbers makes up the order of the coins.
There are thus 30 words, or (n).
Then,
Odd natural number sum = n2.
= 302
= 900
The advisor receives 900 coins in total at the conclusion.
12. From his home, Rahul travels 12 metres north before turning and travelling 35 metres west to his friend's home. He makes his way back by crossing the street diagonally from his friend's home to his own. How far did he travel on his way back?
12. From his home, Rahul travels 12 metres north before turning and travelling 35 metres west to his friend's home. He makes his way back by crossing the street diagonally from his friend's home to his own. How far did he travel on his way back?
Explanation:
Looking up at the figure,
Think about the PQR triangle.
We are aware of Pythagoras' Theorem.
PQ2 + QR2 = PR2
PR2 = 122 + 352
PR2 = 144 + 1225
PR2 = 1369
PR = √1369
PR = 37m
∴Rahul travelled 37 metres to get back to his residence.
13. There were exactly the same number of pupils in each row as there were rows in the lecture hall where 8649 students were seated. In the lecture hall, how many students were seated in each row?
13. There were exactly the same number of pupils in each row as there were rows in the lecture hall where 8649 students were seated. In the lecture hall, how many students were seated in each row?
Explanation:
It is clear from the question that,
There were 8649 pupils in all seated in the lecture hall.
Let's say there are 'a' number of students in each row.
Then,
A is the total number of students.
8649 = a2
Square roots on both sides are taken.
a = √(8649)
a = √(93 × 93)
a = √(93)2
a = 93
There are 93 pupils on each row.
14. A general wants to arrange his 7500 men into a square shape. After planning, he discovered that several of them were overlooked. How many soldiers remained behind?
14. A general wants to arrange his 7500 men into a square shape. After planning, he discovered that several of them were overlooked. How many soldiers remained behind?
Explanation:
The answer to the question is that there are a total of 7500 soldiers.
We can determine the number of soldiers who have been left out by utilising the long division approach.
104 soldiers are missing.
15. The sum of the seats is equal to a.
2704 = a2
15. The sum of the seats is equal to a.
2704 = a2
Explanation:
Square roots on both sides are taken.
a = √(2704)
a = √(52 × 52)
a = √(52)2
a = 52
52 seats are located in each row.
16. A hall can accommodate 2704 people. Find the number of seats in each row if the number of rows equals the number of seats in each row.
16. A hall can accommodate 2704 people. Find the number of seats in each row if the number of rows equals the number of seats in each row.
Explanation:
The total number of seats is listed as 2704 in the question.
Let's say there are 'a' seats in each row.
According to the question's condition,
The number of seats in each row is equal to the number of rows, or 'a'.
Then,
The sum of the seats is equal to a.
2704 = a2
Square roots on both sides are taken.
a = √(2704)
a = √(52 × 52)
a = √(52)2
a = 52
52 seats are located in each row.
17. If 1024 plants are organised so that the number of plants in a row equals the number of rows, determine the number of plants in each row.
17. If 1024 plants are organised so that the number of plants in a row equals the number of rows, determine the number of plants in each row.
Explanation:
It is clear from the question that there are 1024 plants in all.
Let's say there are 'a' number of plants in each row.
According to the question's condition,
There are exactly as many plants in a row as there are rows, or "a."
Then,
A is the total number of plants.
1024 = a2
Square roots on both sides are taken.
a = √(1024)
a = √(32 × 32)
a = √(32)2
a = 32
Each row contains 32 plants in total.
18. Two perfect cubes differ by 189. Find the greater number's cube root if the lesser of the two numbers' cube roots is 3, for example.
18. Two perfect cubes differ by 189. Find the greater number's cube root if the lesser of the two numbers' cube roots is 3, for example.
Explanation:
It is clear from the question that,
Two perfect cubes are different by 189.
The lesser of the two numbers' cube root is equal to three.
So, the cube of the lesser integer is 33.
= 3 × 3 × 3
27
Assume that a3 is the larger number's cube root.
Then,
According to the question's condition,
a3 – 27 = 189
a3 = 189 + 27
a3 =216
By removing the cube's two sides,
a = 3√216
a = 3√(6 × 6 × 6)
a = 3√(63)
a = 6
The greater number's cube root is 6.
19. How many square metres of carpet will be needed to carpet a 6.5 m x 6.5 m square room.
19. How many square metres of carpet will be needed to carpet a 6.5 m x 6.5 m square room.
Explanation:
6.5 metres is the side of the square room.
area of square room thus = 6.52
= 6.5 × 6.5
= 42.25 m2
20. The total of the cubes of three numbers in the ratio 1:2:3 is 4500. Discover the figures.
20. The total of the cubes of three numbers in the ratio 1:2:3 is 4500. Discover the figures.
Explanation:
Assume that the three numbers are a, 2a, and 3a.
Then,
The sum of the cubes of the first three numbers is 4500.
i.e. a3 + (2a)3 + (3a)3 = 4500
a3 + 8a3 + 27a3 = 4500
36a3 = 4500
a3 = 4500/36
a3 = 125
a = 3√125
a = 3√(5 × 5 × 5)
a = 3√(53)
a = 5
The figures are: a = 5.
2a = 2 × 5 = 10
3a = 3 × 5 = 15
21. If the volume of a cube is 512 cm3, determine the length of each side.
21. If the volume of a cube is 512 cm3, determine the length of each side.
Explanation:
It is clear from the question that,
The cube's volume is 512 cm3.
Having said that,
Cube volume = side 3
512 = side3
Cube roots are taken on both sides.
3√512 = side
Side = 3√(8 × 8 × 8)
Side = 3√ (8)3
Side: 8 cm.
The cube has sides that are 8 cm long each.
22. If a square field's perimeter is 96 meters, determine its area.
22. If a square field's perimeter is 96 meters, determine its area.
Explanation:
Given that the square field's perimeter is 96 metres
We are aware that a square's perimeter equals four sides.
Then,
96 = 4 × side
Side = 96/4
Side = 24 m
Therefore, the square's side length is 24 metres.
Now,
Square field area equals (side)2
= 242
= 576 m2
The square field has a 576m2 surface area.
23. 80 metres by 18 metres make up a rectangular field. Calculate the diagonal's length.
23. 80 metres by 18 metres make up a rectangular field. Calculate the diagonal's length.
Explanation:
The question makes it clear that,
80 metres is the length of the rectangle.
Size of rectangular field: 18 metres.
Then
(Length2 + Breadth2) = Diagonal Length
= √(802 + 182)
= √(6400 + 324)
= √6724
= 82 m
∴The diagonal is 82 metres long.
24. Find the volume of a cube whose side measures 15 metres.
24. Find the volume of a cube whose side measures 15 metres.
Explanation:
It is clear from the question that,
One side of a cube is 15 metres long.
Volume of a cube is known to equal (side)3.
= 153
= 15 × 15 × 15
= 3375 m3
∴The cube has a 3375 m3 volume.
25. In a square graph paper with 256 unit squares overall, how many unit squares will there be on each side?
25. In a square graph paper with 256 unit squares overall, how many unit squares will there be on each side?
Explanation:
The question makes it clear that,
256 unit squares make up the entire grid.
Let's assume that the number is "a."
We understand that a a = a2.
So,
a × a = 256
a2 = 256
a = √256
a = √(16 × 16)
a = √162
a = 16
There are 16 unit squares in all.
26. A farmer wants to plough his 150-meter-long square field. How much land will he need to cultivate?
26. A farmer wants to plough his 150-meter-long square field. How much land will he need to cultivate?
Explanation:
It is clear from the question that,
150 metres is the square field's side length.
We are aware that side x side equals a square.
So,
The square field's area is 150 by 150.
= 22,500 m2
The farmer must till 22,500 m2 of land.
27. Find the decimal fraction that results in the number 84.64 when multiplied by itself.
27. Find the decimal fraction that results in the number 84.64 when multiplied by itself.
Explanation:
Assume that the decimal fraction is "a."
We are aware that a a = a2.
So,
a × a = 84.64
a2 = 84.64
a = √84.64
The long division approach is used to determine the required number.
Then,
a = 84.64
a = 9.2
The result of multiplying the decimal fraction 9.2 by itself is 84.64.Find the decimal fraction that results in the number 84.64 when multiplied by itself.
28. One multiplies a decimal number by itself. Find the digit if the product is 51.84.
28. One multiplies a decimal number by itself. Find the digit if the product is 51.84.
Explanation:
Let's assume that the decimal number is "a."
We are aware that a a = a2.
So,
a × a = 51.84
a2 = 51.84
a = √51.84
The long division approach is used to determine the required number.
Then,
a = √51.84
a = 7.2
The result of multiplying the decimal number 7.2 by itself is 51.84.
29. If the diagonal length of a square is 10 cm, determine the length of its side.
29. If the diagonal length of a square is 10 cm, determine the length of its side.
Explanation:
Diagonal length is 10 cm.
Assume ABCD is square.
Sides AB, BC, CD, and DA all equal y.
AC diagonal = 10 cm
Think about triangle ABC now.
According to the Pythagoras principle,
AB2 + BC2 = AC2
102 = y2 + y2
100 = 2y2
y2 = 100/2
y2 = 50
y = √50 cm
y = 5√2 cm
The square's side measures 50 cm (5 inches).
30. The least squares integer that is precisely divisible by 3, 4, 5, 6, and 8 is the answer.
30. The least squares integer that is precisely divisible by 3, 4, 5, 6, and 8 is the answer.
Explanation:
The LCM of 3, 4, 5, 6, and 8 must first be determined.
Therefore, the LCM of 3, 4, 5, 6, and 8 is 2 2 2 3 5.
= 120
The elements are now grouped as follows: (2 + 2) + (2 + 2) + (2 + 3 + 5)
In other words, 2, 3, and 5 are unable to form a pair.
Therefore, 120 must be multiplied by 2 3 5 = 30 in order for it to be a perfect square.
Then,
= 120× 30 = 3600
The least square integer, 3600, is perfectly divided by the numbers 3, 4, 5, 6, and 8.
31. Find the largest three-digit combination that is a square.
31. Find the largest three-digit combination that is a square.
Explanation:
We are aware that 999 is the largest three-digit number.
In order to find a perfect square, we can use the long division approach.
The largest three-digit sum that is a square is 999 minus 38.
= 961
32. The smallest number of four digits that forms a perfect square is to be found.
32. The smallest number of four digits that forms a perfect square is to be found.
Explanation:
We are aware that 1000 is the lowest four-digit number.
In order to find a perfect square, we can use the long division approach
The smallest sum of four numbers that is a square is equal to 1000 plus 24.
= 1024
33. What smallest number needs to be added to 6200 in order for it to be a perfect square?
33. What smallest number needs to be added to 6200 in order for it to be a perfect square?
Explanation:
So, 782 = 6084
When 6084 and 6200 are compared, 6084 6200.
The following perfect square is thus 792 = 6241.
The least number, which should be added to 6200 to get a perfect square, is (6241 - 6200) = 41.
6241 is the necessary perfect square number.
So,
√6241 = √(41 × 41)
= √412
= 41
34. What is the smallest amount that needs to be taken away from 1385 to produce a perfect square? Find the perfect square's square root as well.
34. What is the smallest amount that needs to be taken away from 1385 to produce a perfect square? Find the perfect square's square root as well.
Explanation:
The least amount that may be taken away from 1385 using the long division method to get a perfect square is 16.
Now,
= 1385 – 16
= 1369
So,
√1369 = √(37 × 37)
= √372
= 37
∴Perfect square number 1369 has a square root of 37.
