(i) \(\sqrt[3]{27} + \sqrt[3]{0.008} \) \(+ \sqrt[3]{0.064}\)
By prime factorization of terms,
We have,
= \(\sqrt[3]{27}\)\(+ \sqrt[3]{0.008}\) \(+\sqrt[3]{0.064}\)
= \(\sqrt[3]{3\times3\times3}\) \(+ \sqrt[3]{0.2\times0.02\times0.02}\) \(+ \sqrt[3]{0.04\times0.04\times0.04}\)
= \(\sqrt[3]{3^3} + \sqrt[3]{0.2^3} + \sqrt[3]{0.4^3}\)
= 3 + 0.02 + 0.04
=3.6.
(ii) \(\sqrt[3]{1000} \) \(+ \sqrt[3]{0.008}\) \(- \sqrt[3]{0.125}\)
By prime factorization of terms,
We have,
= \(\sqrt[3]{10\times10\times10}\) \(+ \sqrt[3]{0.2\times0.2\times\times0.2}\) \(-\sqrt[3]{0.05\times0.05\times0.05}\)
= \(\sqrt[3]{10^3} + \sqrt[3]{0.2^3} - \sqrt[3]{0.5^3}\)
= 10 + 0.2 - 0.5
=9.7.
(iii) \(\sqrt[3]{\frac{729}{216}}×\frac{6}{9}\)
By prime factorization of terms,
We have,
= \(\sqrt[3]{\frac{729}{216}}×\frac{6}{9}\)
=\(\sqrt[3]{\frac{9\times9\times9}{6\times6\times6}}\) × \(\frac{6}{9}\)
= \(\frac{\sqrt[3]{9^3}}{\sqrt[3]{6^3}}\times\frac{6}{9}\)
= \(\frac{9}{6}\times\frac{6}{9}\)
=1
(iv) \(\sqrt[3]{\frac{0.027}{0.008}}\)\(÷\sqrt{\frac{0.09}{0.04}} - 1\)
By prime factorization of terms,
We have,
= = \(\sqrt[3]{\frac{0.027}{0.008}}\) \(÷\sqrt{\frac{0.09}{0.04}} \)
= \(\sqrt[3]{\frac{0.3\times0.3\times0.3}{0.2\times0.2\times0.2}}\) ÷ \(\sqrt{\frac{0.3\times0.3\times0.3}{0.2\times0.2}}\)
= \(\frac{\sqrt[3]{0.3^3}}{\sqrt[3]{0.2^3}}\) ÷ \(\frac{\sqrt{0.3^2}}{\sqrt{0.2^2}}\)
= \(\frac{0.3}{0.2}\) ÷ \(\frac{0.3}{0.2}\)
= \(\frac{0.3}{0.2}\)\(\times\frac{0.2}{0.3}\)
= 1.
(v) \(\sqrt[3]{0.1\times0.1\times0.1\times13\times13\times13}\)
By prime factorization of terms, We have,
= \(\sqrt[3]{0.1\times0.1\times0.1\times13\times13\times13}\)
= \(\sqrt[3]{0.1^3\times13^3}\)
= \(\sqrt[3]{0.1^3}\times\) \(\sqrt[3]{13^3}\)
= \(0.1\times13\)
= 1.3.
