Find the cube root of the numbers: $2460375,20346417,210644875,57066625$ using the fact that(i) $2460375 = 3375 \times 729$(ii) $20346417 = 9261 \times 2197$(iii) $210644875 = 42875 \times 4913$(iv) $57066625 = 166375 \times 343$

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To find:

We have to find the cube root of the numbers: $2460375,20346417,210644875,57066625$ using the given facts

Solution:

(i) $\sqrt[3]{2460375}=\sqrt[3]{3375 \times 729}$

$=\sqrt[3]{3375} \times \sqrt[3]{729}$

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

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

$=3 \times 5 \times 3 \times 3$

$=135$

(ii) $\sqrt[3]{20346417}=\sqrt[3]{9261 \times 2197}$

$=\sqrt[3]{9261} \times \sqrt[3]{2197}$

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

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

$=(3 \times 7) \times 13$

$=21 \times 13$

$=273$

(iii) $\sqrt[3]{210644875}=\sqrt[3]{42875 \times 4913}=\sqrt[3]{42875} \times \sqrt[3]{4913}$

$=\sqrt[3]{5 \times 5 \times 5 \times 7 \times 7 \times 7} \times \sqrt[3]{17 \times 17 \times 17}$

$=\sqrt[3]{5^{3} \times 7^{3}} \times \sqrt[3]{17^{3}}$

$=(5 \times 7) \times 17$

$=35 \times 17$

$=595$

(iv) $\sqrt[3]{57066625}=\sqrt[3]{166375 \times 343}$

$=\sqrt[3]{166375} \times \sqrt[3]{343}$

$=\sqrt[3]{5 \times 5 \times 5 \times 11 \times 11 \times 11} \times \sqrt[3]{7 \times 7 \times 7}$

$=\sqrt[3]{5^{3} \times 11^{3}} \times \sqrt[3]{7^{3}}$

$=(5 \times 11) \times 7$

$=55 \times 7$

$=385$

Updated on 10-Oct-2022 13:19:09

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