Ответ:

  1. \(\sqrt2\cdot\sqrt8=\sqrt{16}=4\).
  2. \(\frac{\sqrt2}{\sqrt{18}}=\sqrt{\frac{2}{18}}=\sqrt{\frac19}=\frac13\).
  3. \(\sqrt{27}\cdot\sqrt3=\sqrt{81}=9\).
  4. \(\frac{\sqrt{52}}{\sqrt{117}}=\sqrt{\frac{52}{117}}=\sqrt{\frac49}=\frac23\).
  5. \(\sqrt{28}\cdot\sqrt7=\sqrt{196}=14\).
  6. \(\frac{\sqrt{23}}{\sqrt{2300}}=\sqrt{\frac{23}{2300}}=\sqrt{\frac1{100}}=\frac1{10}\).
  7. \(\sqrt{13}\cdot\sqrt{52}=\sqrt{676}=26\).
  8. \(\frac{\sqrt{12500}}{\sqrt{500}}=\sqrt{25}=5\).

Ответ: 1) 4; 2) \(\frac13\); 3) 9; 4) \(\frac23\); 5) 14; 6) \(\frac1{10}\); 7) 26; 8) 5.