Question

Please solve this, will rate 5 stars and mark as STAR!​

Answer 1

`\boxed{5 \cdot √(2) \cdot \sqrt[6]{5} }`

Explanation:

`\sqrt[3]{250} \cdot \sqrt{\sqrt[3]{10} }`

`\sqrt{\sqrt[3]{10} } \implies (10^(1)/(3) )^(1)/(2) =10^(1)/(6) =\sqrt[6]{10}`

`\therefore \sqrt{\sqrt[3]{10} }=\sqrt[6]{10}`

`\text{Solving }\sqrt[3]{250} \cdot \sqrt{\sqrt[3]{10} }`

`250=2 \cdot 5^3`

`\sqrt[3]{250}=\sqrt[3]{2\cdot 5^3}=5 \sqrt[3]{2}`

Once

`\sqrt[6]{2} \cdot \sqrt[6]{5} = \sqrt[6]{10}`

We have

`5 \sqrt[3]{2} \cdot \sqrt[6]{2} \cdot \sqrt[6]{5}`

We can proceed considering the common base of exponentials

`\sqrt[3]{2} \cdot \sqrt[6]{2} = 2^{(1)/(3)} \cdot 2^{(1)/(6) } = 2^{(3)/(6) } = 2^{(1)/(2) }=√(2)`

Therefore,

`5 \sqrt[3]{2} \cdot \sqrt[6]{2} \cdot \sqrt[6]{5} = 5 \cdot √(2) \cdot \sqrt[6]{5}`