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Find the cube roots of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that(i) 3048625 = 3375 x 729(ii) 20346417 = 9261 x 2197(iii) 210644875 = 42875 x 4913(iv) 57066625 = 166375 x 343 |
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Answer» (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, = \(\sqrt[3]{5\times5\times5\times11\times11\times11}\)\(\times\sqrt[3]{7\times7\times7}\) = \(\sqrt[3]{5^3\times11^3\times}\) \(\sqrt[3]{7^3}\) = \(\sqrt[3]{5^3}\times\sqrt[3]{11^3}\times\) \(\sqrt[3]{7^3}\) = \(5\times7\times11 = 385.\) |
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