To simplify the expression \((\sqrt{0.7} + \sqrt{70})^{2}\), we will follow a step-by-step approach. ### Step 1: Understand the Expression The expression \((\sqrt{0.7} + \sqrt{70})^{2}\) is a binomial squared. The formula for squaring a binomial \((a + b)^{2}\) is given by: \[ (a + b)^{2} = a^{2} + 2ab + b^{2} \] In our case, \(a = \sqrt{0.7}\) and \(b = \sqrt{70}\). ### Step 2: Apply the Formula Using the formula, we can expand the expression: \[ (\sqrt{0.7} + \sqrt{70})^{2} = (\sqrt{0.7})^{2} + 2(\sqrt{0.7})(\sqrt{70}) + (\sqrt{70})^{2} \] ### Step 3: Calculate Each Term 1. **Calculate \((\sqrt{0.7})^{2}\)**: \[ (\sqrt{0.7})^{2} = 0.7 \] 2. **Calculate \((\sqrt{70})^{2}\)**: \[ (\sqrt{70})^{2} = 70 \] 3. **Calculate \(2(\sqrt{0.7})(\sqrt{70})\)**: \[ 2(\sqrt{0.7})(\sqrt{70}) = 2\sqrt{0.7 \times 70} \] Now, calculate \(0.7 \times 70\): \[ 0.7 \times 70 = 49 \] Therefore, \[ 2\sqrt{49} = 2 \times 7 = 14 \] ### Step 4: Combine All Terms Now, we can combine all the calculated terms: \[ 0.7 + 70 + 14 \] Calculating this gives: \[ 0.7 + 70 = 70.7 \] Then, \[ 70.7 + 14 = 84.7 \] ### Final Answer Thus, the simplified expression \((\sqrt{0.7} + \sqrt{70})^{2}\) equals \(84.7\). ### Why the Other Options Are Incorrect - **Option B (70.7)**: This option only accounts for the sum of \(\sqrt{0.7}\) and \(\sqrt{70}\) without squaring the entire expression. It does not include the necessary cross-term \(2(\sqrt{0.7})(\sqrt{70})\). - **Option C (217.7)**: This option is too high and does not reflect the correct calculations. It likely results from a misunderstanding of the squaring process or incorrect addition. - **Option D (168.7)**: Similar to option C, this value is also incorrect and does not correspond to any logical step in the simplification process. ### Revision Summary - Use the binomial expansion formula \((a + b)^{2} = a^{2} + 2ab + b^{2}\) for squaring expressions. - Calculate each term carefully, ensuring to multiply correctly. - Combine all terms accurately to find the final result. - Always double-check calculations to avoid common pitfalls in arithmetic. The correct answer is **A. 84.7**.