Given $\sqrt{2} = 1.414$. The value of $\sqrt{8} + 2\sqrt{32} - 3\sqrt{128} + 4\sqrt{50}$ is:

### Concept & Formula Before substituting a given decimal value, always reduce all radical expressions to their simplest base form. By extracting perfect squares, complex surds will collapse into identical like terms. $$ \sqrt{x^2y} = x\sqrt{y} $$ ### Step-by-Step Solution * Simplify each term individually to isolate $\sqrt{2}$: * $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$ * $2\sqrt{32} = 2\sqrt{16 \times 2} = 2(4\sqrt{2}) = 8\sqrt{2}$ * $3\sqrt{128} = 3\sqrt{64 \times 2} = 3(8\sqrt{2}) = 24\sqrt{2}$ * $4\sqrt{50} = 4\sqrt{25 \times 2} = 4(5\sqrt{2}) = 20\sqrt{2}$ * Substitute these values back into the expression: * $2\sqrt{2} + 8\sqrt{2} - 24\sqrt{2} + 20\sqrt{2}$ * Factor out the common $\sqrt{2}$ and combine coefficients: * $(2 + 8 - 24 + 20)\sqrt{2}$ * $= (30 - 24)\sqrt{2}$ * $= 6\sqrt{2}$ * Substitute the given decimal value ($\sqrt{2} = 1.414$): * $6 \times 1.414 = 8.484$ ### Exam Strategy & Shortcut Look closely at the given clue: $\sqrt{2} = 1.414$. This is a massive hint that every single term in the equation contains a hidden factor of 2. You can immediately divide 8, 32, 128, and 50 by 2 mentally to find the perfect squares (4, 16, 64, 25). Extract their roots directly (2, 4, 8, 5), multiply by their outer coefficients, and sum them up in one fluid mental step to get $6\sqrt{2}$. ### Common Pitfall Substituting the value of $1.414$ too early is a fatal exam mistake. Attempting to calculate $\sqrt{8}$ as $\sqrt{4 \times 1.414}$ or similar bizarre math will completely derail your logic. Always simplify to the very end before converting to decimals. ### Final Answer **Therefore, the correct answer is 8.484.**