Chapter 3 – Indices and Cube Roots
Practice set 3.1
1. Express the following numbers in index form.
(1) Fifth root of 13
Ans: \((13)^{\frac{1}{5}}\)
(2) Sixth root of 9
Ans: \((9)^{\frac{1}{6}}\)
(3) Square root of 256
Ans: \((256)^{\frac{1}{2}}\)
(4) Cube root of 17
Ans: \((17)^{\frac{1}{3}}\)
(5) Eighth root of 100
Ans: \((100)^{\frac{1}{8}}\)
(6) Seventh root of 30
Ans: \((30)^{\frac{1}{7}}\)
2. Write in the form ‘nth root of a’ in each of the following numbers.
(1) \((81)^{\frac{1}{4}}\)
Ans: Fourth root of 81.
(2) \((49)^{\frac{1}{2}}\)
Ans: Square root of 49.
(3) \((15)^{\frac{1}{5}}\)
Ans: Fifth root of 15.
(4) \((512)^{\frac{1}{9}}\)
Ans: Ninth root of 512.
(5) \((100)^{\frac{1}{19}}\)
Ans: Nineteenth root of 100.
(6) \((6)^{\frac{1}{7}}\)
Ans: Seventh root of 6.
Practice set 3.2
1. Complete the following table.
Ans:
2. Write the following numbers in the form of rational indices.
(1) Square root of 5th power of 121.
Ans: \((121)^{\frac{5}{2}}\)
(2) Cube of 4th root of 324
Ans: \((324)^{\frac{4}{3}}\)
(3) 5th root of square of 264
Ans: \((264)^{\frac{5}{2}}\)
(4) Cube of cube root of 3
Ans: \((3)^{\frac{3}{3}}\)
Practice set 3.3
1. Find the cube roots of the following numbers.
(1) 8000
Solution:
8000
= 2 × 2 × 2 × 10 × 10 × 10
= (2 × 10) × (2 × 10) × (2 × 10)
= (2 × 10)³
= 20³
∴ \(\sqrt[3]{8000}\) = 20
(2) 729
Solution:
729
= (3 × 3) × (3 × 3) × (3 × 3)
= (3 × 3)³
= 9³
∴ \(\sqrt[3]{729}\) = 9
(3) 343
Solution:
343
= 7 × 7 × 7
= 7³
∴ ∴ \(\sqrt[3]{343}\) = 7
(4) – 512
Solution:
512
= 2 × 2 × 2 × 4 × 4 × 4
= (2 × 4) × (2 × 4) × (2 × 4)
= (2 × 4)³
= 8³
∴ – 512
= (– 8) × (– 8) × (– 8)
= (– 8)³
∴ \(\sqrt[3]{–\,512}\) = − 8
(5) – 2744
Solution:
2744
= 2 × 2 × 2 × 7 × 7 × 7
= (2 × 7) × (2 × 7) × (2 × 7)
= (2 × 7)³
= 14³
∴ – 2744
= (– 14) × (– 14) × (– 14)
= (– 14)³
∴ ∴ \(\sqrt[3]{–\,2744}\) = − 14
(6) 32768
Solution:
32768
= 2 × 2 × 2 × 4 × 4 × 4 × 4 × 4 × 4
= (2 × 4 × 4) × (2 × 4 × 4) × (2 × 4 × 4)
= (2 × 4 × 4)³
= 32³
∴ \(\sqrt[3]\,–\,32768}\) = \(\large \frac {3}{5}\)
2. Simplify :
(1) \(\sqrt[3]{\frac{27}{125}}\)
Solution:
\(\sqrt[3]\frac{27}{125}}\)
= \(\large {\frac{\sqrt[3]27}{\sqrt[3]125}}\)
= \(\large {\frac{\sqrt[3]3\,×\,3\,×\,3}{\sqrt[3]5\,×\,5\,×\,5}}\)
= \(\large {\frac{\sqrt[3]3³}{\sqrt[3]5³}}\)
= \(\large {\frac{3}{5}\)
∴ \(\sqrt[3]\frac{27}{125}}\) = \(\large {\frac{3}{5}\)
(2) \(\sqrt[3]\frac{16}{54}}\)
Solution:
\(\sqrt[3]\frac{16}{54}}\)
= \(\large {\frac{\sqrt[3]8\,×\,2}{\sqrt[3]27\,×\,2}}\)
= \(\large {\frac{\sqrt[3]8}{\sqrt[3]27}}\)
= \(\large {\frac{\sqrt[3]2\,×\,2\,×\,2}{\sqrt[3]3\,×\,3\,×\,3}}\)
= \(\large {\frac{\sqrt[3]2³}{\sqrt[3]3³}}\)
= \(\large {\frac{2}{3}\)
∴ \(\sqrt[3]\frac{16}{54}}\) = \(\large {\frac{2}{3}\)
3. If \(\sqrt[3]{729}\) = 9 then \(\sqrt[3]{0.000729}\) = ?
Solution:
\(\sqrt[3]{0.000729}\)
= \(\sqrt[3]\frac{729}{1000000}}\)
= \(\large {\frac[3]{9³}{\sqrt[3]100³}}\)
= \(\large {\frac{\sqrt[3]9}{\sqrt[3]100}}\)
= 0.09
∴ \(\sqrt[3]{0.000729}\) = 0.09