Find the cube root of each of the following natural numbers:
(i) 343
(ii) 2744
(iii) 4913
(iv) 1728
(v) 35937
(vi) 17576
(vii) 134217728
(viii) 48228544
(ix) 74088000
(x) 157464
(xi) 1157625
(xii) 33698267.

To do:

We have to find the cube root of the each of the given natural numbers.

Solution:

(i)  Prime factorisation of 343 is,

$343=7 \times 7 \times 7$

$\sqrt[3]{343}=\sqrt[3]{7 \times 7 \times 7}$

$=\sqrt[3]{7^{3}}$

$=7$

(ii)  Prime factorisation of 2744 is,

$2744=2 \times 2 \times 2 \times 7 \times 7 \times 7$

$\sqrt[3]{2744}=\sqrt[3]{2 \times 2 \times 2 \times 7 \times 7 \times 7}$

$=\sqrt[3]{2^{3} \times 7^{3}}$

$=2 \times 7$

$=14$

(iii) Prime factorisation of 4913 is,

$4913=17 \times 17 \times 17$

$\sqrt[3]{4913}=\sqrt[3]{17 \times 17 \times 17}$

$=\sqrt[3]{17^{3}}$

$=17$ 

(iv) Prime factorisation of 1728 is,

$1728=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$

$\sqrt[3]{1728}=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3}$

$=\sqrt[3]{2^{3} \times 2^{3} \times 3^{3}}$

$=2 \times 2 \times 3$

$=12$ 

(v) Prime factorisation of 35937 is,

$35937=3 \times 3 \times 3 \times 11 \times 11 \times 11$

$\sqrt[3]{35937}=\sqrt[3]{3 \times 3 \times 3 \times 11 \times 11 \times 11}$

$=\sqrt[3]{3^{3} \times 11^{3}}$

$=3 \times 11$

$=33$

(vi) Prime factorisation of 17576 is,

$17576=2 \times 2 \times 2 \times 13 \times 13 \times 13$

$\sqrt[3]{17576}=\sqrt[3]{2 \times 2 \times 2 \times 13 \times 13 \times 13}$

$=\sqrt[3]{2^{3} \times 13^{3}}$

$=2 \times 13$

$=26$

(vii) Prime factorisation of 134217728 is,

$134217728=2\times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\times 2 \times 2 \times 2\times2$

$\sqrt[3]{134217728}=\sqrt[3]{2\times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\times 2 \times 2 \times 2\times2}$

$=\sqrt[3]{2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{3} \times 2^{3}}$

$=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

$=512$

(viii) Prime factorisation of 48228544 is,

$48228544=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7 \times 13 \times 13 \times 13$

$\sqrt[3]{48228544}=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 7 \times 13 \times 13 \times 13}$

$=\sqrt[3]{2^{3} \times 2^{3} \times 7^{3} \times 13^{3}}$

$=2 \times 2 \times 7 \times 13$

$=364$

(ix) Prime factorisation of 74088000 is,

$74088000=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7$

$74088000=\sqrt[3]{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7}$

$=\sqrt[3]{2^{3} \times 2^{3} \times 3^{3} \times 5^{3} \times 7^{3}}$

$=2 \times 2 \times 3 \times 5 \times 7$

$=420$

(x) Prime factorisation of 157464 is,

$157464=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$

$\sqrt[3]{157464}=\sqrt[3]{2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3}$

$=\sqrt[3]{2^{3} \times 3^{3} \times 3^{3} \times 3^{3}}$

$=2 \times 3 \times 3 \times 3$

$=54$

(xi) Prime factorisation of 1157625 is,

$1157625=3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7$

$\sqrt[3]{1157625}=\sqrt[3]{3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7}$

$=\sqrt[3]{3^{3} \times 5^{3} \times 7^{3}}$

$=3 \times 5 \times 7$

$=105$

(xii) Prime factorisation of 33698267 is,

$33698267=17 \times 17 \times 17 \times 19 \times 19 \times 19$

$\sqrt[3]{33698267}=\sqrt[3]{17 \times 17 \times 17 \times 19 \times 19 \times 19}$

$=\sqrt[3]{17^{3} \times 19^{3}}$

$=17 \times 19$

$=323$

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Updated on: 10-Oct-2022

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