The square root of $$\frac{(0.75)^3}{1 - 0.75} + [0.75 + (0.75)^2 + 1]$$ is

### Concept & Formula Use the algebraic identity for the difference of two cubes to simplify the fraction: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ $$\frac{a^3}{1 - a} + (a^2 + a + 1) = \frac{a^3 + (1 - a)(a^2 + a + 1)}{1 - a}$$ ### Step-by-Step Solution Let $a = 0.75$. The expression under the square root can be rewritten as: $$\frac{a^3}{1 - a} + (a^2 + a + 1)$$ Find a common denominator to combine the terms: $$\frac{a^3 + (1 - a)(1 + a + a^2)}{1 - a}$$ Expand the numerator using the identity $(1 - a)(1 + a + a^2) = 1^3 - a^3$: $$\frac{a^3 + (1 - a^3)}{1 - a} = \frac{1}{1 - a}$$ Substitute $a = 0.75$ back into the simplified expression: $$\frac{1}{1 - 0.75} = \frac{1}{0.25} = \frac{1}{\frac{1}{4}} = 4$$ The question asks for the square root of this entire value: $$\sqrt{4} = 2$$ ### Exam Strategy & Shortcut Recognize the structure: $\frac{a^3}{1-a} + a^2 + a + 1$. Combining them forms a telescoping algebraic fraction that reduces cleanly to $\frac{1}{1-a}$. With $a = 0.75$, $1-a = 0.25 = 1/4$. The reciprocal is $4$, and its square root is $2$. ### Common Pitfall Students often waste valuable time converting $0.75$ to $3/4$ and manually cubing the decimals. Recognizing the algebraic identity simplifies the entire problem to a single step. ### Final Answer **Therefore, the correct answer is 2.**