Find The cube roots of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that
Question:

Find The cube roots of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that

(i) 3048625 = 3375 × 729

(ii) 20346417 = 9261 × 2197

(iii) 210644875 = 42875 × 4913

(iv) 57066625 = 166375 × 343

Solution:

(i)
To find the cube root, we use the following property:

$\sqrt[3]{a b}=\sqrt[3]{a} \times \sqrt[3]{b}$ for two integers $a$ and $b$

now

$\sqrt[3]{3048625}$

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

$=\sqrt[3]{3375} \times \sqrt[3]{729}$     (By the above property)

$=\sqrt[3]{3 \times 3 \times 3 \times 5 \times 5 \times 5} \times \sqrt[3]{9 \times 9 \times 9} \quad$ (By prime factorisation)

$=\sqrt[3]{\{3 \times 3 \times 3\} \times\{5 \times 5 \times 5\}} \times \sqrt[3]{\{9 \times 9 \times 9\}}$

$=3 \times 5 \times 9$

$=3 \times 5 \times 9$

$=135$

(ii)
To find the cube root, we use the following property:

$\sqrt[3]{a b}=\sqrt[3]{a} \times \sqrt[3]{b}$ for two integers $a$ and $b$

Now,

$\sqrt[3]{20346417}$

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

$=\sqrt[3]{9261} \times \sqrt[3]{2197}$         (By the above property)

$=\sqrt[3]{3 \times 3 \times 3 \times 7 \times 7 \times 7} \times \sqrt[3]{13 \times 13 \times 13}$ (By prime factorisation)

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

$=3 \times 7 \times 13$

$=273$

(iii)
To find the cube root, we use the following property:

$\sqrt[3]{a b}=\sqrt[3]{a} \times \sqrt[3]{b}$ for two integers $a$ and $b$

Now

$\sqrt[3]{210644875}$

$=\sqrt[3]{42875} \times 4913$

$=\sqrt[3]{42875} \times \sqrt[3]{4913}$ (By the above property)

$=\sqrt[3]{5 \times 5 \times 5 \times 7 \times 7 \times 7} \times \sqrt[3]{17 \times 17 \times 17}$     (By prime factorisation)

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

$=5 \times 7 \times 17$

$=595$

(iv)
To find the cube root, we use the following property:

$\sqrt[3]{a b}=\sqrt[3]{a} \times \sqrt[3]{b}$ for two integers $a$ and $b$

now

$\sqrt[3]{57066625}$

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

$=\sqrt[3]{166375} \times \sqrt[3]{343}$     (By the above property)

$=\sqrt[3]{5 \times 5 \times 5 \times 11 \times 11 \times 11} \times \sqrt[3]{7 \times 7 \times 7} \quad$ (By prime factorisation)

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

$=5 \times 11 \times 7$

$=385$