1. Problem: Calculate $\sqrt{0.12} \cdot \sqrt{12}$. Step 1: Use the property of square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Step 2: Multiply inside the root: $\sqrt{0.12 \times 12} = \sqrt{1.44}$. Step 3: Calculate the square root: $\sqrt{1.44} = 1.2$. Answer: $1.2$. 2. Problem: Calculate $(5\sqrt{2} + 1)^2$. Step 1: Use the formula for square of a binomial: $(a+b)^2 = a^2 + 2ab + b^2$. Step 2: Let $a = 5\sqrt{2}$ and $b = 1$. Step 3: Calculate each term: - $a^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$ - $2ab = 2 \times 5\sqrt{2} \times 1 = 10\sqrt{2}$ - $b^2 = 1^2 = 1$ Step 4: Sum all terms: $50 + 10\sqrt{2} + 1 = 51 + 10\sqrt{2}$. Answer: $51 + 10\sqrt{2}$. 3. Problem: Calculate $(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})$. Step 1: Use the difference of squares formula: $(a+b)(a-b) = a^2 - b^2$. Step 2: Let $a = \sqrt{2}$ and $b = \sqrt{7}$. Step 3: Calculate each square: - $a^2 = 2$ - $b^2 = 7$ Step 4: Subtract: $2 - 7 = -5$. Answer: $-5$. 4. Problem: Simplify $\sqrt{294}$. Step 1: Factor 294 into prime factors: $294 = 2 \times 3 \times 7^2$. Step 2: Use the property $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$. Step 3: Extract the perfect square $7^2$: $\sqrt{294} = \sqrt{2 \times 3 \times 7^2} = 7 \sqrt{2 \times 3} = 7 \sqrt{6}$. Answer: $7\sqrt{6}$.