Question

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`\sqrt[3]{3}+√(2)\ \textgreater \ \sqrt[3]{9}\ \textgreater \ \sqrt[4]{13}`

`\;`

`\Large\textrm{Note}`
1. Use of calculator is not allowed.
2. Use of approximated value is not allowed.​

Answer 1

• See below

Explanation:

Use same power to compare the values

Prove the first part:

• 
  `\sqrt[3]{3}+√(2) >\sqrt[3]{9}`
• 
  `\sqrt[6]{3^2}+\sqrt[6]{2^3} >\sqrt[6]{9^2}`
• 
  `\sqrt[6]{9}+\sqrt[6]{8} >\sqrt[6]{81}`
• 
  `\sqrt[6]{9}+\sqrt[6]{8}> \sqrt[6]{8}+\sqrt[6]{8}=2\sqrt[6]{8}=\sqrt[6]{8*64} =\sqrt[6]{512} >\sqrt[6]{81}`
• 
  `512 > 81`

Prove the second part:

• 
  `\sqrt[3]{9} >\sqrt[4]{13}`
• 
  `\sqrt[12]{9^4} >\sqrt[12]{13^3}`
• 
  `\sqrt[12]{6561} >\sqrt[12]{2197}`
• 
  `6561 > 2197`