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#### Page No 15:

#### Question 1:

Express the following numbers in index form.

(1) Fifth root of 13

(2) Sixth root of 9

(3) Square root of 256

(4) Cube root of 17

(5) Eighth root of 100

(6) Seventh root of 30

#### Answer:

It is known that,

*n*^{th} root of *a* is expressed as *a*^{1/n}.

(1) Fifth root of 13

= (13)^{1/5}

(2) Sixth root of 9

= (9)^{1/6}

(3) Square root of 256

= (256)^{1/2}

(4) Cube root of 17

= (17)^{1/3}

(5) Eighth root of 100

= (100)^{1/8}

(6) Seventh root of 30

= (30)^{1/7}

#### Page No 15:

#### Question 2:

Write in the form 'n^{th} root of a' in each of the following numbers.

(1) ${\left(81\right)}^{\frac{1}{4}}$

(2) ${49}^{\frac{1}{2}}$

(3) ${\left(15\right)}^{\frac{1}{5}}$

(4) ${\left(512\right)}^{\frac{1}{9}}$

(5) ${100}^{\frac{1}{19}}$

(6) ${\left(6\right)}^{\frac{1}{7}}$

#### Answer:

It is known that,

*n*^{th} root of *a* is expressed as *a*^{1/n}.

$\left(1\right){\left(81\right)}^{\frac{1}{4}}$

= Fourth root of 81

$\left(2\right){\left(49\right)}^{\frac{1}{2}}$

= Square root of 49

$\left(3\right){\left(15\right)}^{\frac{1}{5}}$

= Fifth root of 15

$\left(4\right){\left(512\right)}^{\frac{1}{9}}$

= Ninth root of 512

$\left(5\right){\left(100\right)}^{\frac{1}{19}}$

= Nineteenth root of 100

$\left(6\right){\left(6\right)}^{\frac{1}{7}}$

= Seventh root of 6

#### Page No 16:

#### Question 1:

Complete the following table.

Sr. No. | Number | Power of the root | Root of the power |

(1) | ${\left(225\right)}^{\frac{3}{2}}$ | Cube of square root of 225 | Square root of cube of 225 |

(2) | ${\left(45\right)}^{\frac{4}{5}}$ | ||

(3) | ${\left(81\right)}^{\frac{6}{7}}$ | ||

(4) | ${\left(100\right)}^{\frac{4}{10}}$ | ||

(5) | ${\left(21\right)}^{\frac{3}{7}}$ |

#### Answer:

Sr. No. | Number | Power of the root | Root of the power |

(1) | ${\left(225\right)}^{\frac{3}{2}}$ | cube of square root of 225 | square root of cube of 225 |

(2) | ${\left(45\right)}^{\frac{4}{5}}$ | 4^{th} power of 5^{th} root of 45 |
5^{th} root of 4^{th} power of 45 |

(3) | ${\left(81\right)}^{\frac{6}{7}}$ | 6^{th} power of 7^{th} root of 81 |
7^{th} root of 6^{th} power of 81 |

(4) | ${\left(100\right)}^{\frac{4}{10}}$ | 4^{th} power of 10^{th} root of 100 |
10^{th} root of 4^{th} power of 100 |

(5) | ${\left(21\right)}^{\frac{3}{7}}$ | cube of 7^{th} root of 21 |
7^{th} root of cube of 21 |

#### Page No 16:

#### Question 2:

Write the following numbers in the form of rational indices.

(1) Square root of 5th power of 121

(2) Cube of 4th root of 324

(3) 5th root of square of 264

(4) Cube of cube root of 3

#### Answer:

It is known that,

*a*^{m/n} = (*a ^{m}*)

^{1/n}means '

*n*

^{th}root of

*m*

^{th}power of

*a*'.

*a*

^{m/n}= (

*a*

^{1/n})

*means '*

^{m}*m*

^{th}power of

*n*

^{th}root of

*a*'.

(1) Square root of 5

^{th}power of 121

$={\left\{{\left(121\right)}^{5}\right\}}^{\frac{1}{2}}\phantom{\rule{0ex}{0ex}}={\left(121\right)}^{\frac{5}{2}}$

(2) Cube of 4

^{th}root of 324

$={\left\{{\left(324\right)}^{\frac{1}{4}}\right\}}^{3}\phantom{\rule{0ex}{0ex}}={\left(324\right)}^{\frac{3}{4}}$

(3) 5

^{th}root of square of 264

$={\left\{{\left(264\right)}^{2}\right\}}^{\frac{1}{5}}\phantom{\rule{0ex}{0ex}}={\left(264\right)}^{\frac{2}{5}}$

(4) Cube of cube root of 3

$={\left\{{3}^{\frac{1}{3}}\right\}}^{3}\phantom{\rule{0ex}{0ex}}={3}^{\frac{3}{3}}$

#### Page No 18:

#### Question 1:

Find the cube roots of the following numbers.

(1) 8000

(2) 729

(3) 343

(4) −512

(5) −2744

(6) 32768

#### Answer:

(1) To find the cube root of 8000, let us factorise 8000 first.

$8000=2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\phantom{\rule{0ex}{0ex}}8000=4\times 4\times 4\times 5\times 5\times 5={4}^{3}\times {5}^{3}={\left(4\times 5\right)}^{3}\left[{a}^{m}\times {b}^{m}={\left(a\times b\right)}^{m}\right]\phantom{\rule{0ex}{0ex}}8000={\left(20\right)}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{8000}=20$

(2) To find the cube root of 729, let us factorise 729 first.

$729=3\times 3\times 3\times 3\times 3\times 3\phantom{\rule{0ex}{0ex}}729=\left(3\times 3\right)\times \left(3\times 3\right)\times \left(3\times 3\right)={\left(3\times 3\right)}^{3}={9}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{729}=9$

(3) To find the cube root of 343, let us factorise 343 first.

$343=7\times 7\times 7={7}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{343}=7$

(4) To find the cube root of −512, let us factorise 512 first.

$512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\phantom{\rule{0ex}{0ex}}512=\left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)\times \left(2\times 2\times 2\right)=8\times 8\times 8\times ={8}^{3}\phantom{\rule{0ex}{0ex}}-512=\left(-8\right)\times \left(-8\right)\times \left(-8\right)={\left(-8\right)}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{-512}=-8$

(5) To find the cube root of −2744, let us factorise 2744 first.

$2744=2\times 2\times 2\times 7\times 7\times 7\phantom{\rule{0ex}{0ex}}2744=\left(2\times 7\right)\times \left(2\times 7\right)\times \left(2\times 7\right)={\left(2\times 7\right)}^{3}={14}^{3}\phantom{\rule{0ex}{0ex}}-2744={\left(-14\right)}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{-2744}=-14$

(6) To find the cube root of 32768, let us factorise 32768 first.

$32768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\phantom{\rule{0ex}{0ex}}32768=\left(2\times 2\times 2\times 2\times 2\right)\times \left(2\times 2\times 2\times 2\times 2\right)\times \left(2\times 2\times 2\times 2\times 2\right)=32\times 32\times 32={\left(32\right)}^{3}\phantom{\rule{0ex}{0ex}}\therefore \sqrt[3]{32768}=32$

#### Page No 18:

#### Question 2:

Simplify:

(1) $\sqrt[3]{\frac{27}{125}}$

(2) $\sqrt[3]{\frac{16}{54}}$

#### Answer:

$\left(1\right)\sqrt[3]{\frac{27}{125}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{3\times 3\times 3}{5\times 5\times 5}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{{3}^{3}}{{5}^{3}}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{{\left(\frac{3}{5}\right)}^{3}}\left[\frac{{a}^{m}}{{b}^{m}}={\left(\frac{a}{b}\right)}^{m}\right]\phantom{\rule{0ex}{0ex}}=\frac{3}{5}$

$\left(2\right)\sqrt[3]{\frac{16}{54}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{2\times 2\times 2\times 2}{2\times 3\times 3\times 3}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{2\times 2\times 2}{3\times 3\times 3}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{\frac{{2}^{3}}{{3}^{3}}}\phantom{\rule{0ex}{0ex}}=\sqrt[3]{{\left(\frac{2}{3}\right)}^{3}}\left[\frac{{a}^{m}}{{b}^{m}}={\left(\frac{a}{b}\right)}^{m}\right]\phantom{\rule{0ex}{0ex}}=\frac{2}{3}$

#### Page No 18:

#### Question 3:

If $\sqrt[3]{729}$ = 9 then $\sqrt[3]{0.000729}$ = ?

#### Answer:

It is given that,

$\sqrt[3]{729}=9\phantom{\rule{0ex}{0ex}}\sqrt[3]{0.000729}=\sqrt[3]{\frac{729}{1000000}}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt[3]{729}}{\sqrt[3]{1000000}}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt[3]{{9}^{3}}}{\sqrt[3]{{\left(100\right)}^{3}}}\phantom{\rule{0ex}{0ex}}=\frac{9}{100}\phantom{\rule{0ex}{0ex}}=0.09$

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