KSEEB Solutions for Class 8 Maths Chapter 5 Squares, Square Roots, Cubes, Cube Roots Ex 5.7

Students can Download Maths Chapter 5 Squares, Square Roots, Cubes, Cube Roots Ex 5.7 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 8 Maths helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

Karnataka Board Class 8 Maths Chapter 5 Squares, Square Roots, Cubes, Cube Roots Ex 5.7

Question 1.
Find the cube root by prime factorisation,
i) 1728
ii) 3375
iii) 10648
iv) 46656
v) 15625
Answer:
i) 1728 = 2 × 2 × 2 × 6 × 6 × 6 = 12 × 12 × 12
\(\sqrt[3]{1728}=12\)

ii) 3375 = 5 × 5 × 5 × 3 × 3 × 3 = 15 × 15 × 15
\(\sqrt[3]{3375}=15\)

iii) 10648 = 2 × 2 × 2 × 11 × 11 × 11
= 2 × 11 × 2 × 11 × 2 × 11
= 22 × 22 × 22
10648 = 22³
∴ \(\sqrt[3]{10648}=22\)

iv) 46656 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
= 4 × 4 × 4 × 9 × 9 × 9
= 4 × 9 × 9 × 4 × 9 × 4
46656 = 36 × 36 × 36 = 36³
∴ \(\sqrt[3]{46656}=36\)

v) 15625 = 5 × 5 × 5 × 5 × 5 × 5
= 25 × 25 × 25
15625 = 25³
∴ \(\sqrt[3]{15625}=25\)

2. Find the cube root of the following by looking at the last digit and using estimation.

Question (i)
91125
Answer:
Unit digit of 91125 in 5. Therefore the units digit in its cube root is 5.
Let us split 91125 as 91 and 125. We find that 43 = 64 < 91 < 125 = 5³
Hence 40³ = 64000 <91125 < 125000 = 50³
∴Cube root of 91125 lies between 40 and 50 and units digit is 5 the only such number is 45.
∴ \(\sqrt[3]{91125}=45\)

Question (ii)
166375
Answer:
Units digit of 166375 is 5. Therefore the units digit of its cube root is 5.
Let us split 166375 as 166 and 375.
5³ = 125 < 166 < 216 = 6³
Hence 50³ = 125000, < 166375 < 216000 = 60³
∴ \(\sqrt[3]{166375}\) lies between 50 and 60. .
Since the units digit is 5 the only such number in 55.
∴ \(\sqrt[3]{166375}=55\)

Question (iii)
704969
Answer:
The units digit of 704969 is 9. Therefore the units digits of its cube root are 9.
Let us split 704969 as 704 and 969
8³ = 512 < 704 < 729 – 9³
Hence = 80³= 512000 < 704969 < 72900 = 90³
∴ \(\sqrt[3]{704969}\) lies between 80 and 90.
Since the units digits is 9 the only such number is 89.
∴ \(\sqrt[3]{704969}=89\)

3. Find the nearest integer to the cube root of each of the following.
(i) 331776
(ii) 46656
(iii) 373248

Question (i)
331776
Answer:
60³ = 216000 < 331776 < 343000 = 70³
Hence \(\sqrt[3]{331776}\) lies between 60 and 70.
We do not know whether 331776 in a perfect cube or not.
However we may sharpen the bound.
68³ = 314432, 69³ = 328509
Hence \(\sqrt[3]{331776}\) lies between 69 and 70
331776 – 328509 = 3267
343000 – 331776 = 11224
331776 in nearer to 69³
∴ The closest integer to \(\sqrt[3]{331776}\) is 69.

Question (ii)
46656
Answer:
30³ = 2700 < 46656 < 64000 – 40₃
\(\sqrt[3]{46656}\) lies between 30 and 40 we do not know whether 46656 in a perfect cube or not. However we may sharper the bound
35³ = 42, 875, 36³ = 46656
∴ \(\sqrt[3]{46656}=36\)

Question (iii)
373248
Answer:
70³ = 343000 < 373248 < 512000 – 80³
\(\sqrt[3]{373248}\) lies between 70 and 80. We do not know whether 373248 is a perfect cube or not.
However we may sharpen the bound. 71³ = 357911, 72³ = 373248
∴ \(\sqrt[3]{373248}=72\)