Let's solve each problem step by step:

1. $\sqrt{\frac{25}{81}}$

• The square root of a fraction can be split: $\sqrt{\frac{25}{81}} = \frac{\sqrt{25}}{\sqrt{81}}$
• $\sqrt{25} = 5$ and $\sqrt{81} = 9$, so: $\frac{5}{9}$

Answer: $\frac{5}{9}$.

3. $\sqrt{\frac{0.49}{4}}$

• Split the square root: $\sqrt{\frac{0.49}{4}} = \frac{\sqrt{0.49}}{\sqrt{4}}$
• $\sqrt{0.49} = 0.7$ (since $0.7 \times 0.7 = 0.49$) and $\sqrt{4} = 2$: $\frac{0.7}{2}$

Answer: $0.35$.

5. $\sqrt{2 \frac{7}{9}}$

• Convert the mixed number $2 \frac{7}{9}$ into an improper fraction: $2 \frac{7}{9} = \frac{25}{9}$
• Now take the square root: $\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$

Answer: $\frac{5}{3}$.

7. $\sqrt{\frac{\frac{1}{16} \cdot \frac{9}{25}}{1}}$

• Multiply the fractions in the numerator: $\frac{1}{16} \cdot \frac{9}{25} = \frac{1 \cdot 9}{16 \cdot 25} = \frac{9}{400}$
• Take the square root: $\sqrt{\frac{9}{400}} = \frac{\sqrt{9}}{\sqrt{400}} = \frac{3}{20}$

Answer: $\frac{3}{20}$.

9. $\sqrt{\frac{0.25 \cdot 49}{9}}$

• Multiply the numbers in the numerator: $0.25 \cdot 49 = 12.25$
• Now the square root becomes: $\sqrt{\frac{12.25}{9}} = \frac{\sqrt{12.25}}{\sqrt{9}}$
• $\sqrt{12.25} = 3.5$ and $\sqrt{9} = 3$: $\frac{3.5}{3} = 1.1667 \quad (\text{or} \, \frac{7}{6})$

Answer: $\frac{7}{6}$.

11. $\sqrt{2 \frac{1}{4} \cdot \frac{1}{16} \cdot \frac{9}{16} \cdot \frac{1}{100}}$

1. Convert $2 \frac{1}{4}$ into an improper fraction: $2 \frac{1}{4} = \frac{9}{4}$
2. Multiply all the fractions: $\frac{9}{4} \cdot \frac{1}{16} \cdot \frac{9}{16} \cdot \frac{1}{100}$
   • Multiply numerators: $9 \cdot 1 \cdot 9 \cdot 1 = 81$
   • Multiply denominators: $4 \cdot 16 \cdot 16 \cdot 100 = 102400$ $\frac{81}{102400}$
3. Take the square root: $\sqrt{\frac{81}{102400}} = \frac{\sqrt{81}}{\sqrt{102400}} = \frac{9}{320}$

Answer: $\frac{9}{320}$.

Summary of Answers:

1. $\frac{5}{9}$
2. $0.35$
3. $\frac{5}{3}$
4. $\frac{3}{20}$
5. $\frac{7}{6}$
6. $\frac{9}{320}$

Do you want me to explain any step further? Let me know!

Related Questions:

1. How do you simplify square roots of fractions with decimals?
2. How do you convert a mixed number to an improper fraction?
3. How do you multiply and simplify multiple fractions?
4. What is the process to find the square root of products of fractions?
5. How do you handle square roots of large denominators?

Tip: Always simplify the problem step by step: fractions first, products second, and then square roots.