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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.​

User Stefan Glienke
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1 Answer

26 votes
26 votes

Answer:

  • 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
User Mikekidder
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