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

Concept & Formula This problem elegantly masks a standard algebraic identity using decimals. By substituting a variable for the decimal, you can reveal the difference of cubes factorization, which perfectly collapses the entire complex expression. $$ a^3 - b^3 = (a-b)(a^2+ab+b^2) $$ Step-by-Step Solution * **Given:** We need to find the square root of $\frac{(0.75)^3}{1-0.75} + [0.75 + (0.75)^2 + 1]$ * **Calculation:** Let $x = 0.75$. Substitute this into the expression: $\frac{x^3}{1-x} + (x + x^2 + 1)$ Rearranging the second part slightly: $= \frac{x^3}{1-x} + (1 + x + x^2)$ * Combine the terms by finding a common denominator, which is $(1-x)$: $= \frac{x^3 + (1-x)(1 + x + x^2)}{1-x}$ * Recognize the algebraic identity in the numerator. The term $(1-x)(1 + x + x^2)$ is exactly the expansion of $(1^3 - x^3)$: $= \frac{x^3 + (1 - x^3)}{1-x}$ * The $x^3$ terms cancel out perfectly: $= \frac{1}{1-x}$ * Substitute $x = 0.75$ back into our simplified expression: $= \frac{1}{1 - 0.75}$ $= \frac{1}{0.25}$ $= \frac{1}{\frac{1}{4}}$ $= 4$ * Finally, the question asks for the **square root** of this entire expression: $\sqrt{4} = 2$ Exam Strategy & Shortcut Whenever you see a fraction with a cube, and a polynomial that looks like $a^2 + a + 1$, immediately think of the difference of cubes identity. Writing out $(0.75)^3$ mathematically is a waste of time. Replace ugly decimals with a variable ($x$) to instantly reveal the underlying algebraic structure. Common Pitfall The most dangerous pitfall is successfully simplifying the massive expression down to 4, and then selecting 4 as the answer (which is an option!). You must carefully read the first three words of the question: "The square root of...". You must take the square root of 4 to finish the problem. Final Answer **Therefore, the correct answer is 2.**