ML Aggarwal

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 4 Cubes and Cube Roots Ex 4.2

Question 1.
Find the cube root of each of the following numbers by prime factorisation:
(i) 12167
(ii) 35937
(iii) 42875
(iv) 21952
(v) 373248
(vi) 32768
(vii) 262144
(viii) 157464
Solution:

Question 2.
Find the cube root of each of the following cube numbers through estimation.
(i) 19683
(ii) 59319
(iii) 85184
(iv) 148877
Solution:

Question 3.
Find the cube root of each of the following numbers:

Solution:

Question 4.
Evaluate the following:
(i) \(\sqrt[3]{512 \times 729}\)
(ii) \(\sqrt[3]{(-1331) \times(3375)}\)
Solution:

Question 5.
Find the cube root of the following decimal numbers:
(i) 0.003375
(ii) 19.683
Solution:

Question 6.
Evaluate: \(\sqrt[3]{27}+\sqrt[3]{0.008}+\sqrt[3]{0.064}\)
Solution:

Question 7.
Multiply 6561 by the smallest number so that product is a perfect cube. Also find the cube root of the product.
Solution:

Question 8.
Divide the number 8748 by the smallest number so that the quotient is a perfect cube. Also, find the cube root of the quotient.
Solution:

Question 9.
The volume of a cubical box is 21952 m³. Find the length of the side of the box.
Solution:

Question 10.
Three numbers are in the ratio 3 : 4 : 5. If their product is 480, find the numbers.
Solution:

Question 11.
Two numbers are in the ratio 4 : 5. If the difference of their cubes is 61, find the numbers.
Solution:

Question 12.
The difference of two perfect cubes is 387. If the cube root of the greater of two numbers is 8, find the cube root of the smaller number.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 4 Cubes and Cube Roots Ex 4.1

Question 1.
Which of the following numbers are not perfect cubes? Give reasons in support of your answer:
(i) 648
(ii) 729
(iii) 8640
(iv) 8000
Solution:

Question 2.
Show that each of the following numbers is a perfect cube. Also, find the number whose cube is the given number:
(i) 1728
(ii) 5832
(iii) 13824
(iv) 35937
Solution:

Question 3.
Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube:
(i) 243
(ii) 3072
(iii) 11979
(iv) 19652
Solution:

Question 4.
Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube:
(i) 1536
(ii) 10985
(iii) 28672
(iv) 13718
Solution:

Question 5.
Rahul makes a cuboid of plasticine of sides 3 cm × 3 cm × 5 cm. How many such cuboids will he need to form a cube?
Solution:

Question 6.
Find the volume of a cubical box whose surface area is 486 cm².
Solution:

Question 7.
Which of the following are cubes of even natural numbers or odd natural numbers:
(i) 125
(ii) 512
(iii) 1000
(iv) 2197
(v) 4096
(vi) 6859
Solution:

Question 8.
Write the ones digit of the cube of each of the following numbers:
(i) 231
(ii) 358
(iii) 419
(iv) 725
(v) 854
(vi) 987
(vii) 752
(viii) 893
Solution:

Question 9.
Find the cubes of the following numbers:
(i) 13
(ii) \(3 \frac{1}{5}\)
(iii) \(-5 \frac{1}{7}\)
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Check Your Progress

Question 1.
Show that 1089 is a perfect square. Also, find the number whose square is 1089.
Solution:

Question 2.
Find the smallest number which should be multiplied by 3675 to make it a perfect square. Also, find the square root of this perfect square.
Solution:

Question 3.
Express 121 as the sum of 11 odd numbers.
Solution:

Question 4.
How many numbers lie between 99² and 100²?
Solution:

Question 5.
Write a Pythagorean triplet whose one number is 17.
Solution:

Question 6.
Find the smallest square number which is divisible by each of the numbers 6, 8, 9.
Solution:

Question 7.
In an auditorium, the number of rows is equal to a number of chairs in each row. If the capacity of the auditorium is 1764. Find the number of chairs in each row.
Solution:

Question 8.
Find the length of diagonal of a rectangle whose length and breadth are 12 m and 5 m respectively.
Solution:

Question 9.
Find the square root of 144 by successive subtraction.
Solution:

Question 10.
Find the square root of the following numbers by prime factorisation:
(i) 5625
(ii) 1521
Solution:

Question 11.
Find the square root of the following numbers by long division method:
(i) 21904
(ii) 108241
Solution:

Question 12.
Find the square root of following decimal numbers:
(i) 17.64
(ii) 13.3225
Solution:

Question 13.
Find the square root of the following fractions:

Solution:

Question 14.
Find the least number which must be subtracted from 2311 to make it a perfect square.
Solution:

Question 15.
Find the least number which must be added to 520 to make it a perfect square.
Solution:

Question 16.
Find the greatest number of 5 digits which is a perfect square.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) A number ending in ……… is never a perfect square.
(ii) On combining two consecutive triangular number, we get a ………
(iii) If a number has digits ……… in the unit’s place, then its square ends in 1.
(iv) Sum of the first 10 odd natural numbers is ………
(v) A number of non-square numbers between 11² and 12² is ………
(vi) Number of zeros in the end of the square of 400 is ………
(vii) Square of any ……… number can be expressed as the sum of two consecutive natural numbers.
(viii)For a natural number m > 1, (2m, m² – 1, m² + 1) is called ………
Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) All-natural numbers are not perfect squares.
(ii) A perfect square can never be expressed as the product of pairs of equal prime factors.
(iii) A number having 2,3,7 or 8 at its unit place is never a square number.
(iv) A number having 0, 1, 4, 5, 6 or 9 at its unit place is always a square number.
(v) A number ending in an even number of zeros is always a perfect square.
(vi) Square of an odd number is always an odd number.
(vii) 1, 3, 6, 10, 15, are called triangular numbers.
(viii)There are 2n non-square numbers between the squares of consecutive numbers n and (n + 1).
(ix) (4, 6, 8) is a Pythagorean triplet.
Solution:

Multiple Choice Questions
Choose the correct answer from the given four options (3 to 15):
Question 3.
How many natural numbers lie between 25² and 26²?
(a) 49
(b) 50
(c) 51
(d) 52
Solution:

Question 4.
Square of an even number is always
(a) even
(b) odd
(c) even or odd
(d) none of these
Solution:

Question 5.
1+ 3 + 5 + 7 + ……….. up to n terms is equal to
(a) n² – 1
(b) (n + 1)²
(c) n² + 1
(d) n²
Solution:

Question 6.
\(\sqrt{208+\sqrt{2304}}\) is equal to
(a) 18
(b) 16
(c) 14
(d) 22
Solution:

Question 7.
\(\sqrt{0.0016}\) is equal to
(a) 0.04
(b) 0.004
(c) 0.4
(d) none of these
Solution:

Question 8.
The smallest number by which 75 should be divided to make it a perfect square is
(a) 1
(b) 2
(c) 3
(d) 4
Solution:

Question 9.
\(\sqrt{3 \frac{6}{25}}\) is equal to

Solution:

Question 10.
The smallest number by which 162 should be multiplied to make it a perfect square is
(a) 4
(b) 3
(c) 2
(d) 1
Solution:

Question 11.
If the area of a square field is 961 unit², then the length of its side is
(a) 29 units
(b) 41 units
(c) 31 untis
(d) 39 units
Solution:

Question 12.
The smallest number that should be subtracted from 300 to make it a perfect square is
(a) 11
(b) 12
(c) 13
(d) 14
Solution:

Question 13.
If one number of the Pythagorean triplet is 6, then the triplet is
(a) (4, 5, 6)
(b) (5, 6, 7)
(c) (6, 7, 8)
(d) (6, 8, 10)
Solution:

Question 14.
nth triangular number is

Solution:

Question 15.
Given that \(\sqrt{1521}\) = 39, the value of \(\sqrt{0.1521}+\sqrt{15.21}\) is
(a) 42.9
(b) 4.29
(c) 3.51
(d) 35.1
Solution:

Value Based Questions
Question 1.
In a school, students of class VIII collected ₹9216 to give a donation to an NGO working for the education of poor children. If each student donated as many rupees as the number of students in class VIII. Find the number of students in class VIII.
Why should we donate money for the education of poor children? What values are being promoted?
Solution:

Question 2.
A person wants to plant 2704 medicinal plants with a board depicting the diseases in which that can be used. He planted these in the form of rows. If each row contains as many plants as the number of rows, then find the number of rows.
Why should we plant medicinal plants? What values are being promoted?
Solution:

Higher Order Thinking Skills (Hots)
Question 1.
A square field is to be ploughed. Ramu get it ploughed in ₹34560 at the rate of ₹15 per sq. m. Find the length of side of square field.
Solution:

Question 2.
Lalit has some chocolates. He distributed these chocolates among 13 children in such a way that he gave one chocolate to first child, 3 chocolates to the second child, 5 chocolates to third and so on. Find the number of chocolates Lalit had.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Ex 3.4

Question 1.
Find the square root of each of the following by division method:
(i) 2401
(ii) 4489
(iii) 106929
(iv) 167281
(v) 53824
(vi) 213444
Solution:

Question 2.
Find the number of digits in the square root of each of the following (without any calculation):
(i) 81
(ii) 169
(iii) 4761
(iv) 27889
(v) 525625
Solution:

Question 3.
Find the square root of the following decimal numbers by division method:
(i) 51.84
(ii) 42.25
(iii) 18.4041
(iv) 5.774409
Solution:

Question 4.
Find the square root of the following numbers correct to two decimal places:
(i) 645.8
(ii) 107.45
(iii) 5.462
(iv) 2
(v) 3
Solution:

Question 5.
Find the square root of the following fractions by division method:

Solution:

Question 6.
Find the least number which must be subtracted from each of the following numbers to make them a perfect square. Also find the square root of the perfect square number so obtained:
(i) 2000
(ii) 984
(iii) 8934
(iv) 11021
Solution:

Question 7.
Find the least number which must be added to each of the following numbers to make them a perfect square. Also find the square root of the perfect square number so obtained:
(i) 1750
(ii) 6412
(iii) 6598
(iv) 8000
Solution:

Question 8.
Find the smallest four-digit number which is a perfect square.
Solution:

Question 9.
Find the greatest number of six digits which is a perfect square.
Solution:

Question 10.
In a right triangle ABC, ∠B = 90°.
(i) If AB = 14 cm, BC = 48 cm, find AC.
(ii) If AC = 37 cm, BC = 35 cm, find AB.
Solution:

Question 11.
A gardener has 1400 plants. He wants to plant these in such a way that the number of rows and number of columns remains the same. Find the minimum number of plants he needs more for this.
Solution:

Question 12.
There are 1000 children in a school. For a P.T. drill they have to stand in such a way that the number of rows is equal to a number of columns. How many children would be left out in this arrangement?
Solution:

Question 13.
Amit walks 16 m south from his house and turns east to walk 63 ra to reach his friend’s house. While returning, he walks diagonally from his friend’s house to reach back to his house. What distance did he walk while returning?
Solution:

Question 14.
A ladder 6 m long leaned against a wall. The ladder reaches the wall to a height of 4.8 m. Find the distance between the wall and the foot of the ladder.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Ex 3.3

Question 1.
By repeated subtraction of odd numbers starting from 1, find whether the following numbers are perfect squares or not? If the number is a perfect square then find its square root:
(i) 121
(ii) 55
(iii) 36
(iv) 90
Solution:

Question 2.
Find the square roots of the following numbers by prime factorisation method:
(i) 784
(ii) 441
(iii) 1849
(iv) 4356
(v) 6241
(vi) 8836
(vii) 8281
(viii) 9025
Solution:

Question 3.
Find the square roots of the following numbers by prime factorisation method:

Solution:

Question 4.
For each of the following numbers, find the smallest natural number by which it should be multiplied so as to get a perfect square. Also, find the square root of the square number so obtained:
(i) 588
(ii) 720
(iii) 2178
(iv) 3042
(v) 6300
Solution:

Question 5.
For each of the following numbers, find the smallest natural number by which it should be divided so that this quotient is a perfect square. Also, find the square root of the square number so obtained:
(i) 1872
(ii) 2592
(iii) 3380
(iv) 16224
(v) 61347
Solution:

Question 6.
Find the smallest square number that is divisible by each of the following numbers:
(i) 3, 6, 10, 15
(ii) 6, 9, 27, 36
(iii) 4, 7, 8, 16
Solution:

Question 7.
4225 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.
Solution:

Question 8.
The area of a rectangle is 1936 sq. m. If the length of the rectangle is 4 times its breadth, find the dimensions of the rectangle.
Solution:

Question 9.
In a school a P.T. teacher wants to arrange 2000 students in the form of rows and columns for P.T. display. If the number of rows is equal to number of columns and 64 students could not be accommodated in this arrangement. Find the number of rows.
Solution:

Question 10.
In a school, the students of class VIII collected ₹2304 for a picnic. Each student contributed as mdny rupees as the number of students in the class. Find the number of students in the class.
Solution:

Question 11.
The product of two numbers is 7260. If one number is 15 times the other number, find the numbers.
Solution:

Question 12.
Find three positive numbers in the ratio 2 : 3 : 5, the sum of whose squares is 950.
Solution:

Question 13.
The perimeter of two squares is 60 metres and 144 metres respectively. Find the perimeter of another square equal in area to the sum of the first two squares.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Ex 3.2

Question 1.
Write five numbers which you can decide by looking at their one’s digit that they are not square numbers.
Solution:

Question 2.
What will be the unit digit of the squares of the following numbers?
(i) 951
(ii) 502
(iii) 329
(iv) 643
(v) 5124
(vi) 7625
(vii) 68327
(viii) 95628
(ix) 99880
(x) 12796
Solution:

Question 3.
The following numbers are obviously not perfect. Give reason.
(i) 567
(ii) 2453
(iii) 5298
(iv) 46292
(v) 74000
Solution:

Question 4.
The square of which of the following numbers would be an odd number or an even number? Why?
(i) 573
(ii) 4096
(iii) 8267
(iv) 37916
Solution:

Question 5.
How many natural numbers lie between the square of the following numbers?
(i) 12 and 13
(ii) 90 and 91
Solution:

Question 6.
Without adding, find the sum.
(i) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15
(ii) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29
Solution:

Question 7.
(i) Express 64 as the sum of 8 odd numbers.
(ii) 121 as the sum of 11 odd numbers.
Solution:

Question 8.
Express the following as the sum of two consecutive integers:
(i) 19²
(ii) 33²
(iii) 47²
Solution:

Question 9.
Find the squares of the following numbers without actual multiplication:
(i) 31
(ii) 42
(iii) 86
(iv) 94
Solution:

Question 10.
Find the squares of the following numbers containing 5 in unit’s place:
(i) 45
(ii) 305
(iii) 525
Solution:

Question 11.
Write a Pythagorean triplet whose one number is
(i) 8
(ii) 15
(iii) 63
(iv) 80
Solution:

Question 12.
Observe the following pattern and find the missing digits:
21² = 441
201² = 40401
2001² = 4004001
20001² = 4 – – – 4 – – – 1
200001² = ————–
Solution:

Question 13.
Observe the following pattern and find the missing digits:
9² = 81
99² = 9801
999² = 998001
9999² = 99980001
99999² = 9——–8———01
999999² = 9——–0———1
Solution:

Question 14.
Observe the following pattern and find the missing digits:
7² = 49
67² = 4489
667² = 444889
6667² = 44448889
66667² = 4 ———–8 ————– 9
666667² = 4———–8————8 –
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 3 Squares and Square Roots Ex 3.1

Question 1.
Which of the following natural numbers are perfect squares? Give reasons in support of your answer.
(i) 729
(ii) 5488
(iii) 1024
(iv) 243
Solution:

Question 2.
Show that each of the following numbers is a perfect square. Also, find the number whose square is the given number.
(i) 1296
(ii) 1764
(iii) 3025
(iv) 3969
Solution:

Question 3.
Find the smallest natural number by which 1008 should be multiplied to make it a perfect square.
Solution:

Question 4.
Find the smallest natural number by which 5808 should be divided to make it a perfect square. Also, find the number whose square is the resulting number.
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 2 Exponents and Powers Check Your Progress

Question 1.
Evaluate:

Solution:

Question 2.
Simplify:

Solution:

Question 3.
Find the multiplicative inverse of \(\left(\frac{125}{27}\right)^{\frac{-2}{3}}\)
Solution:

Question 4.
Simplify and write in exponential form:

Solution:

Question 5.
Simplify: \(\frac{5^{n+2}-6+5^{n+1}}{13 \times 5^{n}-2 \times 5^{n+1}}\)
Solution:

Question 6.
Prove that (a + b)^-1 (a^-1 + b^-1) = (ab)^-1.
Solution:

Question 7.
Find the value of x for which

Solution:

Question 8.
If \(\frac{2^{-n} \times 8^{2 n+1} \times 16^{2 n}}{4^{3 n}}=\frac{1}{16}\), find the value of n.
Solution:

Question 9.
By what number should \(\left[\left(\frac{-5}{2}\right)^{3}\right]^{-3}\) be multiplied to get \(\left(\frac{-2}{5}\right)^{5}\)?
Solution:

Question 10.
By what number should \(\left(\frac{-3}{2}\right)^{-4}\) be divided to get \(\left(\frac{-2}{3}\right)^{3}\)?
Solution:

Question 11.
Express the following numbers in standard form:
(i) 0.0000000003904
(ii) 12730000000000
Solution:

Question 12.
Express the following numbers in the usual form:
(i) 5.73 × 10^-11
(ii) 3.895 × 10¹⁵.
Solution:

Question 13.
If \(\left(\frac{9}{4}\right)^{-4} \times\left(\frac{2}{3}\right)^{3}=\left(\frac{p}{q}\right)^{11}\), then find the value of \(\left(\frac{p}{q}\right)^{-2}\).
Solution:

Question 14.
Mass of the Earth is 5.97 × 10²⁴ kg and mass of the Moon is 7.35 × 10²² kg. What is the difference of their masses?
Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths

ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 2 Exponents and Powers Objective Type Questions

Mental Maths

Question 1.
Fill in the blanks:
(i) Multiplicative inverse of \(\left( \frac { -4 }{ 5 } \right) ^{ 2 }\) = ……..
(ii) The standard form of 0.000182 = ………
(iii) The standard form of ……….. is 0.00000003572.
(iv) Expanded form of 273.92 = ………..
(v) If (-7)^x × \(\left( \frac { 1 }{ -7 } \right) ^{ -3 }\) = 7⁸, then value of x is …….
(vi) If 3ⁿ = 729, then n = ……
(vii) The value of (7⁰ + 5⁰) is ……..
Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) The value of (-3)^-2 is \(\frac { 1 }{ -9 }\).
(ii) For negative integers m and n, a^m × aⁿ = a^m+n (a ≠ 0).
(iii) 3⁰ × 4⁰ = 12⁰.
(iv) The value of 3⁷ – 2⁷ is 1⁷.
(v) The standard form of 0.0034 is 3.4 × 10³.
(vi) Multiplicative inverse of (-2)⁵ is \(\frac { 1 }{ -32 }\).
Solution:

Multiple Choice Questions

Choose the correct answer from the given four options (3 to 12):
Question 3.
\(\left( \frac { 1 }{ 2 } \right) ^{ -5 }\) is equal to
(a) 32
(b) \(\frac { 1 }{ 32 }\)
(c) -32
(d) \(\frac { -1 }{ 32 }\)
Solution:

Question 4.

Solution:

Question 5.
If 2⁴ × 4³ = 4^x, then the value of x is
(a) 5
(b) 7
(c) 4
(d) -5
Solution:

Question 6.

Solution:

Question 7.

Solution:

Question 8.
Standard form of 0.000000789 is
(a) 78.9 × 10^-8
(b) 789 × 10^-9
(c) 7.89 × 10^-7
(d) None
Solution:

Question 9.
The usual form of 3.67 × 10⁵ is
(a) 3670000
(b) 367000
(c) 36700
(d) 3670
Solution:

Question 10.
Multiplicative inverse of \(\left( \frac { -2 }{ 3 } \right) ^{ 4 }\) is
(a) \(\frac { 16 }{ 81 }\)
(b) \(\frac { -16 }{ 81 }\)
(c) \(\frac { 81 }{ 16 }\)
(d) \(\frac { -81 }{ 16 }\)
Solution:

Question 11.

Solution:

Question 12.

Solution:

Value Based Questions

Question 1.
Anaemia causes due to deficiency of Red Blood cells. The size of a red blood cell is 0.000000007 mm. Express it in standard form. Rahul is suffering from anemia, his doctor adviced him to take healthy food regularly and avoid junk food.
Why should we eat healthy food? What values are being promoted?
Solution:

Higher Order Thinking Skills (HOTS)

Question 1.
Find the multiplicative inverse of (5⁰ + 3⁰) (5⁰ – 3⁰).
Solution:

Question 2.

Solution:

Question 3.

Solution:

ML Aggarwal Class 8 Solutions for ICSE Maths