Indices & Cube Root
Maharashtra Board-Class-8-Mathematics-Chapter-3
Solution
Practice Set 3.1
Practice Set 3.1
Question 1.
Express the following numbers in index form.
(1) Fifth root of 13
\(13^{\frac{1}{5}}\)
(2) Sixth root of 9
\(9^{\frac{1}{6}}\)
(3) Square root of 256
\(256^{\frac{1}{2}}\)
(4) Cube root of 17
\(17^{\frac{1}{3}}\)
(5) Eighth root of 100
\(100^{\frac{1}{8}}\)
(6) Seventh root of 30
\(30^{\frac{1}{7}}\)
Question 2.
Write in the form ‘nth root of a’ in each of the following numbers.
(1) \(81^{\frac{1}{4}}\)
Fourth root of 81
(2) \(49^{\frac{1}{2}}\)
Square root of 49.
(3) \(15^{\frac{1}{5}}\)
Fifth root of 15
(4) \(512^{\frac{1}{9}}\)
Ninth root of 512
(5) \(100^{\frac{1}{19}}\)
Nineteenth root of 100
(6) \(6^{\frac{1}{7}}\)
Seventh root of 6.
Practice Set 3.2
Question 1.
Complete the following table.
Sr. No. | Number | Power of the root | Root of the power |
1 | \(225^{\frac{3}{2}}\) | Cube of square root of 225 | Square root of cube of 225 |
2 | \(45^{\frac{4}{5}}\) | ||
3 | \(81^{\frac{6}{7}}\) | ||
4 | \(100^{\frac{4}{10}}\) | ||
5 | \(21^{\frac{3}{7}}\) |
Sr. No. | Number | Power of the root | Root of the power |
1 | \(225^{\frac{3}{2}}\) | Cube of square root of 225 | Square root of cube of 225 |
2 | \(45^{\frac{4}{5}}\) | Fourth power of fifth root of 45 | 5th root of 4th power of 45 |
3 | \(81^{\frac{6}{7}}\) | Sixth power of 7th root of 81 | 7th root of 6th power of 81 |
4 | \(100^{\frac{4}{10}}\) | Fourth power of 10th root of 100 | 10th root of 4th power of 100 |
5 | \(21^{\frac{3}{7}}\) | Third power of 7th root of 21 | 7th root of 3rd power of 21 |
Question 2.
Write the following numbers in the form of rational indices.
(1) Square root of 5th power of 121.
\((121)^{\frac{5}{2}}\)
(2) Cube of 4th root of 324
\((324)^{\frac{3}{4}}\)
(3) 5th root of square of 264
\((264)^{\frac{2}{5}}\)
(4) Cube of cube root of 3
\((3)^{\frac{3}{3}}\)
Practice Set 3.3
Question 1.
Find the cube roots of the following numbers.
(1) 8000
8000 :
= 8 x 1000
= 2 x 2 x 2 x 10 x 10 x 10
= 2^{3} x 10^{3}
= (2 x 10)^{3}
= 20^{3}
Answer is \(\sqrt[3]{8000}\) = 20
(2) 729
729 :
= 9 x 81
= 9 x 9 x 9
= 9^{3}
Answer is \(\sqrt[3]{729}\) = 9
(3) 343
343 :
= 7 x 49
= 7 x 7 x 7
= 7^{3}
Answer is \(\sqrt[3]{343}\) = 7
(4) −512
-512 :
First factorise 512
= 2 x 256
= 2 x 2 x 128
= 2 x 2 x 2 x 64
= 2 x 2 x 2 x 4 x 16
= 2 x 2 x 2 x 4 x 4 x 4
= 2^{3} x 4^{3} = (2 x 4)^{3}
∴ − 512 = (−8)^{3}
Answer is \(\sqrt[3]{-512}\) = -8
(5) −2744
2744 is divisible by 2
∴ it must be divisible by 8
2744 :
= 8 x 343
= 8 x 7 x 49
= 2 x 2 x 2 x 7 x 7 x 7
= 2^{3} x 7^{3}
= (2x7)^{ 3}
= 14^{3}
Answer is \(\sqrt[3]{-2744}\)= -14
(6) 32768
= 8 x 4096
= 8 x 8 x 512
= 8 x 8 x 8 x 64
= 8 x 8 x 8 x 4 x 4 x 4
= 8^{3} x 4^{3}
= (8 x 4)^{3}
= 32^{3}
Answer is \(\sqrt[3]{32768}\) = 32
Question 2.
Simplify :
(1) \(\sqrt[3]{\frac{27}{125}}\)
\(\frac{27}{125}\) = \(\frac{3×3×3}{5×5×5}=\frac{3^3}{5^3}\)
∴ \(\sqrt[3]{\frac{27}{125}}\)=\(\sqrt[3]{\frac{3^3}{5^3}}=\frac{3}{5}\)
Answer is \(\sqrt[3]{\frac{27}{125}}\)=\(\frac{3}{5}\)
(2) \(\sqrt[3]{\frac{16}{54}}\)
\(\frac{16}{54}\) = \(\frac{8×2}{27×2}=\frac{2^3}{3^3}\)
∴ \(\sqrt[3]{\frac{16}{54}}\)=\(\sqrt[3]{\frac{2^3}{3^3}}=\frac{2}{3}\)
Answer is \(\sqrt[3]{\frac{16}{54}}\)=\(\frac{2}{3}\)
(3) If \(\sqrt[3]{729}\) = 9 then = \(\sqrt[3]{0.000729}\) ?
\(\sqrt[3]{0.000729}\)=\(\sqrt[3]{\frac{729}{1000000}}\)
\(\sqrt[3]{\frac{729}{1000000}}\)=\(\frac{9}{100}\)= 0.09
Answer is \(\sqrt[3]{\frac{729}{1000000}}\)= 0.09
Main Page : – Maharashtra Board Class 8th Mathematics – All chapters notes, solutions, videos, test, pdf.
Notes : Chapter 3 : Indices and Cube root
Next Chapter : Chapter 4– –online Solution
We reply to valid query.