use rational expression to write (see comments for work) as a single radical expression

\[\sqrt[4]{2}\times \sqrt[3]{5}\]

Just remember that in a fractional (rational) exponent the top number is the power and the bottom number is the root. so: \[\sqrt[4]{2}=2^{\frac{1}{4}}\]

Now you do the third root of 5

the options are \[\sqrt[12]{10} , \sqrt[12]{10000000} , \sqrt[12]{5000}, 1/5000^{12}\]

The third one.

\[2^{\frac{1}{4}}(5^{\frac{1}{3}})=x\] \[\log2^{\frac{1}{4}}(5^{\frac{1}{3}})=\log x\] \[\log 2^{\frac{1}{4}}+\log 5^{\frac{1}{3}}=logx\] \[\frac{1}{4}\log2 + \frac{1}{3}\log5 = \log x\] \[\frac{3}{12}\log2 + \frac{4}{12} \log 5 = \log x\] \[\frac{1}{12}(3\log 2 + 4 \log 5 ) = \log x\] \[\frac{1}{12}(\log 2^3 + \log 5^4)=\log x\] \[\frac{1}{12}(\log 8 + \log 625)=\log x\] \[\frac{1}{12} \log 5000=\log x\] \[\log 5000^{\frac{1}{12}}=\log x\] \[5000^{\frac{1}{12}}=x\] \[\sqrt[12]{5000}=x\]

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