Punjab State Board PSEB 8th Class Maths Book Solutions Chapter 7 Cubes and Cube Roots Ex 7.2 Textbook Exercise Questions and Answers.
PSEB Solutions for Class 8 Maths Chapter 7 Cubes and Cube Roots Ex 7.2
1. Find the cube root of each of the following numbers by prime factorisation method.
Question (i).
64
Solution:
\(\begin{array}{l|l}
2 & 64 \\
\hline 2 & 32 \\
\hline 2 & 16 \\
\hline 2 & 8 \\
\hline 2 & 4 \\
\hline 2 & 2 \\
\hline & 1
\end{array}\)
By prime factorisation,
64 = 2 × 2 × 2 × 2 × 2 × 2
∴ \(\sqrt[3]{64}\) = 2 × 2
= 4
Thus, cube root of 64 is 4.
Question (ii).
512
Solution:
\(\begin{array}{l|l}
2 & 512 \\
\hline 2 & 256 \\
\hline 2 & 128 \\
\hline 2 & 64 \\
\hline 2 & 32 \\
\hline 2 & 16 \\
\hline 2 & 8 \\
\hline 2 & 4 \\
\hline 2 & 2 \\
\hline & 1
\end{array}\)
By prime factorisation,
512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
∴ \(\sqrt[3]{512}\) = 2 × 2 × 2
= 8
Thus, cube root of 512 is 8.
Question (iii).
10648
Solution:
\(\begin{array}{r|l}
2 & 10648 \\
\hline 2 & 5324 \\
\hline 2 & 2662 \\
\hline 11 & 1331 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}\)
By prime factorisation,
10648 = 2 × 2 × 2 × 11 × 11 × 11
∴ \(\sqrt[3]{10648}\) = 2 × 11
= 22
Thus, cube root of 10648 is 22.
Question (iv).
27000
Solution:
\(\begin{array}{l|l}
2 & 27000 \\
\hline 2 & 13500 \\
\hline 2 & 6750 \\
\hline 3 & 3375 \\
\hline 3 & 1125 \\
\hline 3 & 375 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
By prime factorisation
27000 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5
∴ \(\sqrt[3]{27000}\) = 2 × 3 × 5
= 30
Thus, cube root of 27000 is 30.
Question (v).
15625
Solution:
\(\begin{array}{l|l}
5 & 15625 \\
\hline 5 & 3125 \\
\hline 5 & 625 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
By prime factorisation,
15625 = 5 × 5 × 5 × 5 × 5 × 5
∴ \(\sqrt[3]{15625}\) = 5 × 5
= 25
Thus, cube root of 15625 is 25.
Question (vi).
13824
Solution:
\(\begin{array}{l|l}
2 & 13824 \\
\hline 2 & 6912 \\
\hline 2 & 3456 \\
\hline 2 & 1728 \\
\hline 2 & 864 \\
\hline 2 & 432 \\
\hline 2 & 216 \\
\hline 2 & 108 \\
\hline 2 & 54 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
By prime factorisation,
13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
∴ \(\sqrt[3]{13824}\) = 2 × 2 × 2 × 3
= 24
Question (vii).
110592
Solution:
\(\begin{array}{l|l}
2 & 110592 \\
\hline 2 & 55296 \\
\hline 2 & 27648 \\
\hline 2 & 13824 \\
\hline 2 & 6912 \\
\hline 2 & 3456 \\
\hline 2 & 1728 \\
\hline 2 & 864 \\
\hline 2 & 432 \\
\hline 2 & 216 \\
\hline 2 & 108 \\
\hline 2 & 54 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
By prime factorisation,
110592 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
∴ \(\sqrt[3]{110592}\) = 2 × 2 × 2 × 2 × 3
= 48
Thus, cube root of 110592 is 48.
Question (viii).
46656
Solution:
\(\begin{array}{l|l}
2 & 46656 \\
\hline 2 & 23328 \\
\hline 2 & 11664 \\
\hline 2 & 5832 \\
\hline 2 & 2916 \\
\hline 2 & 1458 \\
\hline 3 & 729 \\
\hline 3 & 243 \\
\hline 3 & 81 \\
\hline 3 & 27 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}\)
By prime factorisation,
46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
∴ \(\sqrt[3]{46656}\) = 2 × 2 × 3 × 3
= 36
Thus, cube root of 46656 is 36.
Question (ix).
175616
Solution:
\(\begin{array}{l|l}
2 & 175616 \\
\hline 2 & 87808 \\
\hline 2 & 43904 \\
\hline 2 & 21952 \\
\hline 2 & 10976 \\
\hline 2 & 5488 \\
\hline 2 & 2744 \\
\hline 2 & 1372 \\
\hline 2 & 686 \\
\hline 7 & 343 \\
\hline 7 & 49 \\
\hline 7 & 7 \\
\hline & 1
\end{array}\)
By prime factorisation,
175616 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 7
∴ \(\sqrt[3]{175616}\) = 2 × 2 × 2 × 7
= 56
Thus, cube root of 175616 is 56.
Question (x).
91125
Solution:
\(\begin{array}{l|l}
3 & 91125 \\
\hline 3 & 30375 \\
\hline 3 & 10125 \\
\hline 3 & 3375 \\
\hline 3 & 1125 \\
\hline 3 & 375 \\
\hline 5 & 125 \\
\hline 5 & 25 \\
\hline 5 & 5 \\
\hline & 1
\end{array}\)
By prime factorisation,
91125 = 3 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5
∴ \(\sqrt[3]{91125}\) =3 × 3 × 5
= 45
Thus, cube root of 91125 is 45.
2. State true or false.
Question (i).
Cube of any odd number is even.
Solution:
False, cube of any odd number is odd.
Question (ii).
A perfect cube does not end with two zeros.
Solution:
True, a perfect cube ending with zero will always have a triplet of zeros.
Question (iii).
If square of a number ends with 5, then its cube ends with 25.
Solution:
False, let us understand with an example.
152 = 15 × 15 = 225
153 = 15 × 15 × 15 = 3375
Question (iv).
There is no perfect cube which ends with 8.
Solution:
False, let us understand with an example.
8 = 23, 1728 = 123
Question (v).
The cube of a two-digit number may be a three-digit number.
Solution:
False, let us take an example.
The smallest two-digit number is 10.
103 = 1000
which is a four-digit number and not a three-digit number.
Question (vi).
The cube of a two-digit number may have seven or more digits.
Solution:
False, let us take an example.
The greatest two-digit number is 99.
993 = 970299
which is a six-digit number.
Question (vii).
The cube of a single digit number may be a single digit number.
Solution:
True, let us take examples.
13 = 1 and 23 = 8
3. You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
Solution:
Yes, we can guess without prime factorisation.
1331 : Separate given number into two groups.
1331 → 1 and 331
331 → Units place digit of 331 is 1.
∴ Unit place digit of cube root of 1331 = 1 (∵ 13 = 1)
1 → 13 = 1 and 23 = 8
∴ Tens digit of cube root of 1331 = 1
Thus, the cube root of 1331 is 11.
(i) 4913 : Separate given number into two groups.
4913 → 4 and 913
913 → Units place digit of 913 is 3.
∴ Unit digit of cube root of 4913 = 7 (∵ 73 = 343)
4 → 13 = 1 and 23 = 8
1 < 4 < 8 (∴ 13 < 4 < 23)
∴ The tens digit of cube root of 4913 = 1
Thus, the cube root of 4913 is 17.
(ii) 12167 : Separate given number into two groups.
12167 → 12 and 167
167 → Units place digit of 167 is 7.
∴ Unit digit of cube root of 12167 = 3
(∵ 33 = 27)
12 → 23 = 8 and 33 = 27
8 < 12 < 27 (∵ 23 < 12 < 33)
∴ Tens place digit of cube root of 12167 = 2
Thus, the cube root of 12167 is 23.
(iii) 32768 : Separate given number into two groups.
32768 → 32 and 768
768 → Units place digit of 768 is 8.
∴ Unit digit of cube root of 32768 = 2 (∵ 23 = 8)
32 → 33 = 27 and 43 = 64
27 < 32 < 64 (∵ 33 < 32 < 43)
∴ The tens digit of cube root of 32768 = 3
Thus, the cube root of 32768 is 32.