(i) 3048625 = 3375 x 729
Taking cube root of the whole, we get,
= \(\sqrt[3]{3048625}\) = \(\sqrt[3]{3375\times729}\)
We know that,
= \(\sqrt[3]{ab}\) = \(\sqrt[3]{a\times}\) \(\sqrt[3]{b}\)
= \(\sqrt[3]{3048625}\) = \(\sqrt[3]{3375}\) x \(\sqrt[3]{729}\)
Now by prime factorization,
= \(\sqrt[3]{3\times3\times3\times5\times5\times5}\) x \(\sqrt[3]{9\times9\times9}\)
= \(\sqrt[3]{3^3\times5^3}\times\sqrt[3]{9^3}\)
= \(\sqrt[3]{3^3}\times\) \(\sqrt[3]{5^3}\times\sqrt[3]{9^3}\)
= \(3\times5\times9 = 135.\)
(ii) 20346417 = 9261 x 2197
Taking cube root of the whole,
= \(\sqrt[3]{20346417}\) = \(\sqrt[3]{9261\times2197}\)
We know that,
= \(\sqrt[3]{ab}\) = \(\sqrt[3]{a\times}\) \(\sqrt[3]{b}\)
= \(\sqrt[3]{9261\times2197}\) = \(\sqrt[3]{9261}\times\) \(\sqrt[3]{2197}\)
Now by prime factorization,
= \(\sqrt[3]{3\times3\times3\times7\times7\times7}\) \(\times\sqrt[3]{13\times13\times13}\)
= \(\sqrt[3]{3^3\times7^3}\times\) \(\sqrt[3]{13^3}\)
= \(\sqrt[3]{3^3}\times\) \(\sqrt[3]{7^3}\times\) \(\sqrt[3]{13^3}\)
= 3 × 7 × 13 = 273.
(iii) 210644875 = 42875 x 4913
Taking cube root of the whole,
= \(\sqrt[3]{210644875}\) = \(\sqrt[3]{42875\times4913}\)
We know that,
= \(\sqrt[3]{ab}\) = \(\sqrt[3]{a\times}\) \(\sqrt[3]{b}\)
= \(\sqrt[3]{42875\times4913}\) = \(\sqrt[3]{42875\times}\) \(\sqrt[3]{4913}\)
Now by prime factorization,
= \(\sqrt[3]{5\times5\times5\times7\times7\times7}\) x \(\sqrt[3]{17\times17\times17}\)
= \(\sqrt[3]{5^3\times7^3}\times\) \(\sqrt[3]{13^3}\)
= \(\sqrt[3]{5^3}\times\) \(\sqrt[3]{7^3}\times\) \(\sqrt[3]{17^3}\)
= \(5\times7\times17 = 595.\)
(iv) 57066625 = 166375 x 343
Taking cube root of the whole, we get,
= \(\sqrt[3]{57066625}\) = \(\sqrt[3]{166375\times343}\)
We know that,
= \(\sqrt[3]{ab}\) = \(\sqrt[3]{a\times}\) \(\sqrt[3]{b}\)
Now by prime factorization method,
= \(\sqrt[3]{5\times5\times5\times11\times11\times11}\)\(\times\sqrt[3]{7\times7\times7}\)
= \(\sqrt[3]{5^3\times11^3\times}\) \(\sqrt[3]{7^3}\)
= \(\sqrt[3]{5^3}\times\sqrt[3]{11^3}\times\) \(\sqrt[3]{7^3}\)
= \(5\times7\times11 = 385.\)
