The problem in the image asks to evaluate the following expression:

$\sqrt{1 - \frac{1}{4}} \cdot \left( \sqrt{16 + \frac{1}{3}} - \sqrt{9 - \frac{2}{3}} \right) = ?$

Let’s solve this step by step:

Step 1: Simplify the first square root:

$\sqrt{1 - \frac{1}{4}} = \sqrt{\frac{4}{4} - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$

Step 2: Simplify the second square root:

$\sqrt{16 + \frac{1}{3}} = \sqrt{\frac{48}{3} + \frac{1}{3}} = \sqrt{\frac{49}{3}} = \frac{\sqrt{49}}{\sqrt{3}} = \frac{7}{\sqrt{3}} = \frac{7\sqrt{3}}{3}$

Step 3: Simplify the third square root:

$\sqrt{9 - \frac{2}{3}} = \sqrt{\frac{27}{3} - \frac{2}{3}} = \sqrt{\frac{25}{3}} = \frac{\sqrt{25}}{\sqrt{3}} = \frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Step 4: Subtract the square roots from Step 2 and Step 3:

$\frac{7\sqrt{3}}{3} - \frac{5\sqrt{3}}{3} = \frac{(7 - 5)\sqrt{3}}{3} = \frac{2\sqrt{3}}{3}$

Step 5: Multiply the result by the first term:

$\frac{\sqrt{3}}{2} \cdot \frac{2\sqrt{3}}{3} = \frac{\sqrt{3} \cdot 2\sqrt{3}}{2 \cdot 3} = \frac{2 \cdot 3}{6} = 1$

Thus, the result of the expression is 1.

The correct answer is C) 1.

Would you like more details or have any other questions?

Related Questions:

1. How do you simplify square roots with fractions inside?
2. How can you quickly subtract terms with square roots in the numerator?
3. What are some common mistakes in handling nested radicals?
4. How can this expression be approached using approximate values for square roots?
5. How can this problem be adapted to use variables instead of numbers?

Tip:

Always simplify individual components (like square roots and fractions) before combining terms in more complex expressions.