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

Answer 1

(i)
To find the cube root, we use the following property:
$\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}$ for two integers a and b

Now

$$\begin{gathered} \sqrt[3]{3048625} \\ =\sqrt[3]{3375\times 729} \end{gathered}$$
$=\sqrt[3]{3375}\times \sqrt[3]{729}$ (By the above property)
$=\sqrt[3]{3\times 3\times 3\times 5\times 5\times 5}\times \sqrt[3]{9\times 9\times 9}$ (By prime factorisation)
$$\begin{gathered} =\sqrt[3]{\left\{3\times 3\times 3\right\}\times \left\{5\times 5\times 5\right\}}\times \sqrt[3]{\left\{9\times 9\times 9\right\}} \\ =3\times 5\times 9 \\ =135 \end{gathered}$$

Thus, the answer is 135.

(ii)
To find the cube root, we use the following property:
$\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}$ for two integers a and b

Now

$$\begin{gathered} \sqrt[3]{20346417} \\ =\sqrt[3]{9261\times 2197} \end{gathered}$$
$=\sqrt[3]{9261}\times \sqrt[3]{2197}$ (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)
$$\begin{gathered} =\sqrt[3]{\left\{3\times 3\times 3\right\}\times \left\{7\times 7\times 7\right\}}\times \sqrt[3]{\left\{13\times 13\times 13\right\}} \\ =3\times 7\times 13 \\ =273 \end{gathered}$$

Thus, the answer is 273.

(iii)
To find the cube root, we use the following property:
$\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}$ for two integers a and b

Now

$$\begin{gathered} \sqrt[3]{210644875} \\ =\sqrt[3]{42875\times 4913} \end{gathered}$$
$=\sqrt[3]{42875}\times \sqrt[3]{4913}$ (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)
$$\begin{gathered} =\sqrt[3]{\left\{5\times 5\times 5\right\}\times \left\{7\times 7\times 7\right\}}\times \sqrt[3]{\left\{17\times 17\times 17\right\}} \\ =5\times 7\times 17 \\ =595 \end{gathered}$$

Thus, the answer is 595.

(iv)
To find the cube root, we use the following property:
$\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}$ for two integers a and b

Now

$$\begin{gathered} \sqrt[3]{57066625} \\ =\sqrt[3]{166375\times 343} \end{gathered}$$
$=\sqrt[3]{166375}\times \sqrt[3]{343}$ (By the above property)
$=\sqrt[3]{5\times 5\times 5\times 11\times 11\times 11}\times \sqrt[3]{7\times 7\times 7}$ (By prime factorisation)
$$\begin{gathered} =\sqrt[3]{\left\{5\times 5\times 5\right\}\times \left\{11\times 11\times 11\right\}}\times \sqrt[3]{\left\{7\times 7\times 7\right\}} \\ =5\times 11\times 7 \\ =385 \end{gathered}$$

Thus, the answer is 385.