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 = 23 x 103 = (2 x 10)3 = 203 Answer is \(\sqrt[3]{8000}\) = 20
(2) 729
729 : = 9 x 81 = 9 x 9 x 9 = 93 Answer is \(\sqrt[3]{729}\) = 9
(3) 343
343 : = 7 x 49 = 7 x 7 x 7 = 73 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 = 23 x 43 = (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 = 23 x 73 = (2x7) 3 = 143 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 = 83 x 43 = (8 x 4)3 = 323 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
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