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1.	Evaluate:(i) \(\sqrt[3]{4^3\times6^3}\)(ii) \(\sqrt[3]{8\times17\times17\times17}\)(iii) \(\sqrt[3]{700\times2\times49\times5}\)(iv) \(125\sqrt[3]{a^3}\) - \(\sqrt[3]{125a^6}\)
Answer» (i) \(\sqrt[3]{4^3\times6^3}\) We have, = \(\sqrt[3]{4^3\times6^3}\) = \(\sqrt[3]{4^3}\times\) \(\sqrt[3]{6^3}\) = \(4\times6\) = 24. (ii) \(\sqrt[3]{8\times17\times17\times17}\) We have, = \(\sqrt[3]{8\times17\times17\times17}\) = \(\sqrt[3]{8\times}\) \(\sqrt[3]{17^3}\) = \(\sqrt[3]{2^3}\times\) \(\sqrt[3]{17^3}\) = \(2\times17 \) = 34. (iii) \(\sqrt[3]{700\times2\times49\times5}\) We have, = \(\sqrt[3]{700\times2\times49\times5}\) Getting prime factors of numbers, = \(\sqrt[3]{700\times2\times49\times5}\) = \(\sqrt[3]{2\times2\times2\times5\times5\times7\times7\times5}\) = \(\sqrt[3]{2^3\times5^3\times7^3}\) = \(\sqrt[3]{2^3}\times\)\(\sqrt[3]{5^3}\times\) \(\sqrt[3]{7^3}\) = 2 × 5 × 7 = 70. (iv) \(125\sqrt[3]{a^3}\) - \(\sqrt[3]{125a^6}\) We have, = \(125\sqrt[3]{a^6}\) - \(\sqrt[3]{125a^6}\) = \(125\sqrt[3]{(a^2) ^3}\) - \(\sqrt[3]{5^3(a^2)^3}\) = \(125a^2\) - \(\sqrt[3]{5^3(a^2)^3}\) = \(125a^2\) - \(\sqrt[3]{5^3}\times\sqrt[3]{(a^2)^3}\) = \(125a^2\) - \(5a^2\) = \(120a^2. \)

Evaluate:(i) \(\sqrt[3]{4^3\times6^3}\)(ii) \(\sqrt[3]{8\times17\times17\times17}\)(iii) \(\sqrt[3]{700\times2\times49\times5}\)(iv) \(125\sqrt[3]{a^3}\) - \(\sqrt[3]{125a^6}\)

Answer»

(i) \(\sqrt[3]{4^3\times6^3}\)

We have,

= \(\sqrt[3]{4^3\times6^3}\)

= \(\sqrt[3]{4^3}\times\) \(\sqrt[3]{6^3}\)

= \(4\times6\)

= 24.

(ii) \(\sqrt[3]{8\times17\times17\times17}\)

We have,

= \(\sqrt[3]{8\times17\times17\times17}\)

= \(\sqrt[3]{8\times}\) \(\sqrt[3]{17^3}\)

= \(\sqrt[3]{2^3}\times\) \(\sqrt[3]{17^3}\)

= \(2\times17 \)

= 34.

(iii) \(\sqrt[3]{700\times2\times49\times5}\)

We have,

= \(\sqrt[3]{700\times2\times49\times5}\)

Getting prime factors of numbers,

= \(\sqrt[3]{700\times2\times49\times5}\)

= \(\sqrt[3]{2\times2\times2\times5\times5\times7\times7\times5}\)

= \(\sqrt[3]{2^3\times5^3\times7^3}\)

= \(\sqrt[3]{2^3}\times\)\(\sqrt[3]{5^3}\times\) \(\sqrt[3]{7^3}\)

= 2 × 5 × 7 = 70.

(iv) \(125\sqrt[3]{a^3}\) - \(\sqrt[3]{125a^6}\)

We have,

= \(125\sqrt[3]{a^6}\) - \(\sqrt[3]{125a^6}\)

= \(125\sqrt[3]{(a^2) ^3}\) - \(\sqrt[3]{5^3(a^2)^3}\)

= \(125a^2\) - \(\sqrt[3]{5^3(a^2)^3}\)

= \(125a^2\) - \(\sqrt[3]{5^3}\times\sqrt[3]{(a^2)^3}\)

= \(125a^2\) - \(5a^2\)

= \(120a^2. \)

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