Let's break down and solve the mathematical expression in the image step by step:

$\frac{1}{3} \cdot \sqrt{0.09} + 2 \cdot \frac{1}{7} \cdot \sqrt{0.0196} + \sqrt{576} \cdot 0.\overline{6}$

Step 1: Simplify the square roots.

• $\sqrt{0.09} = 0.3$
• $\sqrt{0.0196} = 0.14$
• $\sqrt{576} = 24$

Step 2: Multiply by the corresponding fractions and constants.

• $\frac{1}{3} \cdot 0.3 = 0.1$
• $2 \cdot \frac{1}{7} \cdot 0.14 = 2 \cdot 0.02 = 0.04$
• $24 \cdot 0.\overline{6} = 24 \cdot \frac{2}{3} = 16$

Step 3: Add the terms together.

$0.1 + 0.04 + 16 = 16.14$

Thus, the value of the expression is 16.14.

Would you like more details on any step or have further questions?

Related Questions:

1. How do you calculate square roots of decimal numbers?
2. How are repeating decimals, like $0.\overline{6}$, converted into fractions?
3. How do you multiply fractions with decimals?
4. What is the general formula for simplifying an expression involving square roots and fractions?
5. How do you handle mixed operations like addition and multiplication in complex expressions?

Tip:

When dealing with repeating decimals, they can be converted into fractions for easier manipulation in calculations. For example, $0.\overline{6} = \frac{2}{3}$.