The square root of $(272^2 - 128^2)$ is

### Concept & Formula Avoid squaring large individual numbers directly. Instead, apply the difference of squares algebraic identity to simplify the internal expression. Difference of squares identity: $$ a^2 - b^2 = (a - b)(a + b) $$ ### Step-by-Step Solution - **Step 1: Set up the targeted expression.** - We need to compute: $\sqrt{272^2 - 128^2}$ - **Step 2: Apply the algebraic identity.** - Let $a = 272$ and $b = 128$. - $272^2 - 128^2 = (272 - 128)(272 + 128)$ - **Step 3: Compute values within brackets.** - $272 - 128 = 144$ - $272 + 128 = 400$ - So, the expression becomes $\sqrt{144 \times 400}$ - **Step 4: Factor out the square roots.** - $\sqrt{144 \times 400} = \sqrt{144} \times \sqrt{400}$ - $12 \times 20 = 240$ ### Exam Strategy & Shortcut Never square $272$ and $128$ manually. The difference of squares structure turns a messy subtraction problem into a product of two obvious perfect squares ($\sqrt{144 \times 400}$). Taking roots gives $12 \times 20 = 240$ instantly. ### Common Pitfall Mistakenly applying the distributive error $\sqrt{a^2 - b^2} = a - b$, which results in computing $272 - 128 = 144$. Radical distribution over subtraction is completely invalid. ### Final Answer Therefore, the correct answer is 240.