$$ \left[ \frac{8 (3.75)^3 + 1}{(7.5)^2 - 6.5} \right] $$ is equal to:

### Concept & Formula This is a heavily disguised sum of cubes problem. The challenge is using the properties of exponents to manipulate the numerator and denominator into the standard $a^3 + b^3$ identity format. The core formulas are: $$ (xy)^n = x^n y^n $$ $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution Let's resolve the numerator first to reveal the true value of our variable $a$. We know that $8 = 2^3$. Rewrite the first term in the numerator: $$ 8(3.75)^3 = 2^3 \times (3.75)^3 $$ Using exponent rules, combine the bases: $$ (2 \times 3.75)^3 = (7.5)^3 $$ So, the numerator is actually $(7.5)^3 + 1^3$. This tells us that $a = 7.5$ and $b = 1$. Now, let's verify if the denominator matches the $a^2 - ab + b^2$ pattern required for the shortcut: $$ a^2 - ab + b^2 = (7.5)^2 - (7.5 \times 1) + 1^2 $$ $$ = (7.5)^2 - 7.5 + 1 $$ $$ = (7.5)^2 - 6.5 $$ The calculated pattern perfectly matches the given denominator! The entire expression represents: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ Which simplifies to $(a + b)$. Calculate the final result: $$ 7.5 + 1 = 8.5 $$ ### Exam Strategy & Shortcut Test setters love masking numbers. Whenever you see a coefficient like $8$ or $27$ next to a cubed term, instantly pull it inside the parentheses as its cube root ($2$ or $3$). Doing $2 \times 3.75 = 7.5$ immediately reveals the hidden variable $a$. From there, $7.5 + 1 = 8.5$ is a straightforward mental step. ### Common Pitfall Attempting to compute $3.75^3$ manually or trying to expand $7.5^2$. This question is designed specifically to punish brute-force arithmetic. If a calculation looks impossibly tedious, step back—you are missing an algebraic substitution. ### Final Answer **Therefore, the correct answer is 8.5.**