Here is the breakdown of the problems from the image. I will solve them one by one. Let me know where you'd like detailed explanations.

Solutions

10. $\frac{5^{-19}}{(5^3)^{-6}}$

Simplify the denominator:
$(5^3)^{-6} = 5^{-18}.$
Thus:
$\frac{5^{-19}}{5^{-18}} = 5^{-19 - (-18)} = 5^{-1} = \frac{1}{5}.$

11. $\sqrt{128} \cdot \sqrt{2}$

Combine under a single square root:
$\sqrt{128 \cdot 2} = \sqrt{256} = 16.$

12. $\sqrt{29.4} \cdot \sqrt{60}$

Combine under one square root:
$\sqrt{29.4 \cdot 60} = \sqrt{1764}.$
Simplify:
$\sqrt{1764} = 42.$

13. $\frac{\sqrt{175}}{\sqrt{7}}$

Simplify using division inside the square root:
$\frac{\sqrt{175}}{\sqrt{7}} = \sqrt{\frac{175}{7}} = \sqrt{25} = 5.$

14. $\frac{8\sqrt{108}}{\sqrt{3}}$

Simplify the square root:
$\frac{\sqrt{108}}{\sqrt{3}} = \sqrt{\frac{108}{3}} = \sqrt{36} = 6.$
Multiply:
$8 \cdot 6 = 48.$

15. $\frac{\sqrt{147}}{4\sqrt{3}}$

Simplify the numerator:
$\sqrt{147} = \sqrt{49 \cdot 3} = 7\sqrt{3}.$
Thus:
$\frac{\sqrt{147}}{4\sqrt{3}} = \frac{7\sqrt{3}}{4\sqrt{3}} = \frac{7}{4}.$

16. $\frac{(9\sqrt{11})^2}{18}$

Square the numerator:
$(9\sqrt{11})^2 = 81 \cdot 11 = 891.$
Divide:
$\frac{891}{18} = 49.$

17. $\sqrt{36} \cdot 92$

Simplify the square root:
$\sqrt{36} = 6.$
Multiply:
$6 \cdot 92 = 552.$

18. $\frac{2}{3} \sqrt{54} \cdot \sqrt{6}$

Combine the square roots:
$\sqrt{54} \cdot \sqrt{6} = \sqrt{54 \cdot 6} = \sqrt{324} = 18.$
Multiply:
$\frac{2}{3} \cdot 18 = 12.$

19. $(\sqrt{14} - \sqrt{17})(\sqrt{14} + \sqrt{17})$

This is a difference of squares:
$a^2 - b^2 = (\sqrt{14})^2 - (\sqrt{17})^2 = 14 - 17 = -3.$

20. $(\sqrt{50} + \sqrt{128}) \cdot 2\sqrt{2}$

Simplify each square root:
$\sqrt{50} = 5\sqrt{2}, \quad \sqrt{128} = 8\sqrt{2}.$
Add:
$\sqrt{50} + \sqrt{128} = 5\sqrt{2} + 8\sqrt{2} = 13\sqrt{2}.$
Multiply by $2\sqrt{2}$:
$13\sqrt{2} \cdot 2\sqrt{2} = 13 \cdot 2 \cdot 2 = 52.$

Final Answers

1. $\frac{1}{5}$
2. $16$
3. $42$
4. $5$
5. $48$
6. $\frac{7}{4}$
7. $49$
8. $552$
9. $12$
10. $-3$
11. $52$

Let me know if you'd like detailed steps for any specific problem!

Related Questions:

1. How do you simplify square roots with multiplication or division?
2. What is the rule for simplifying exponents in fractions?
3. How can you identify and solve a difference of squares?
4. How do you simplify complex square root expressions?
5. How do you handle rationalizing denominators with square roots?

Tip: Always combine square roots into one expression before simplifying. This reduces errors and makes calculations easier!