Answer:
The product of [tex](\sqrt[3]{4})(\sqrt{2})(\sqrt[6]{8})[/tex] is [tex]2\sqrt[3]{4}[/tex].
Step-by-step explanation:
- Transform the radicand into similar bases.
[tex](\sqrt[3]{4})(\sqrt{2})(\sqrt[6]{8})=(\sqrt[3]{2^{2} })(\sqrt{2})(\sqrt[6]{2^{3}})[/tex] - Change the following from radicals to exponents. Note that when radicals turn into exponents, the radicand's exponent is the numerator, while the index or the nth root is the denominator.
[tex](\sqrt[3]{2^{2} })(\sqrt{2})(\sqrt[6]{2^{3}})\\(2^{\frac{2}{3}})(2^{\frac{1}{2}})(2^{\frac{3}{6} })[/tex]
3/6 when simplified is 1/2.
[tex](2^{\frac{2}{3}})(2^{\frac{1}{2}})(2^{\frac{3}{6} })\\(2^{\frac{2}{3}})(2^{\frac{1}{2}})(2^{\frac{1}{2} })[/tex] - By the product law (laws of exponent), [tex]a^{m}*a^{n} =a^{m+n}[/tex].
[tex](2^{\frac{2}{3}})(2^{\frac{1}{2}})(2^{\frac{1}{2} })\\2^{\frac{2}{3}+\frac{1}{2}+\frac{1}{2}}\\ 2^{\frac{2}{3}+1 } \\2^{1\frac{2}{3} } =2^{\frac{5}{3} }[/tex] - You can now transform your answer back into a radical form.
[tex]2^{\frac{5}{3} } =\sqrt[3]{2^{5} }[/tex] - Simplify
[tex]\sqrt[3]{2^{5} }\\ \sqrt[3]{32}\\ \sqrt[3]{(8)(4)} \\2\sqrt[3]{4}[/tex]
