<s>[INST] How to show that some nested radicals simplify to other Show that $\sqrt{8}\cdot\sqrt{9-\sqrt{77}}=2\cdot\sqrt{11}-2\cdot\sqrt{7}$
I have tried multiplying the radicals but that didn't work.
The resulting radicals do not add up or get subtracted.
I have tried taking commons also but that also doesn't work.
 [/INST] multiply both side of the fraction to $\sqrt {2}$
$$\sqrt { 8 } \cdot \sqrt { 9-\sqrt { 77 }  } =\sqrt { 8 } \frac { \sqrt { 18-2\sqrt { 77 }  }  }{ \sqrt { 2 }  } =\sqrt { 8 } \frac { \sqrt { { \left( \sqrt { 7 } -\sqrt { 11 }  \right)  }^{ 2 } }  }{ \sqrt { 2 }  } =\frac { \sqrt { 8 }  }{ \sqrt { 2 }  } \left| \sqrt { 7 } -\sqrt { 11 }  \right| =2\left( \sqrt { 11 } -\sqrt { 7 }  \right) $$
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