The value of $$ \left( \frac{0.125 + 0.027}{0.5 \times 0.5 + 0.09 - 0.15} \right) $$ is

### Concept & Formula This problem hides a classic algebraic identity behind raw decimal numbers. The core concept is recognizing perfect cubes and using the sum of cubes expansion. The required formula is: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution Observe the numbers in the numerator. They are perfect cubes: * $0.125 = 0.5 \times 0.5 \times 0.5 = (0.5)^3$ * $0.027 = 0.3 \times 0.3 \times 0.3 = (0.3)^3$ Let $a = 0.5$ and $b = 0.3$. The numerator translates to $a^3 + b^3$. Now examine the denominator to see if it matches the $a^2 - ab + b^2$ pattern: * $0.5 \times 0.5$ represents $a^2$. * $0.09$ is exactly $(0.3)^2$, which represents $b^2$. * $0.15$ is exactly $0.5 \times 0.3$, which represents $ab$. The denominator perfectly matches the form $a^2 + b^2 - ab$. Write the full expression algebraically: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ Expand the numerator using the identity: $$ \frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2} $$ The quadratic term cancels out from the top and bottom, leaving only $(a + b)$. Substitute the values back: $$ 0.5 + 0.3 = 0.8 $$ ### Exam Strategy & Shortcut When you spot cubes like $0.125$ and $0.027$ in the numerator over a mixed quadratic form, instantly extract the cube roots: $0.5$ and $0.3$. If the sign in the numerator is positive, simply add the roots: $0.5 + 0.3 = 0.8$. You can solve this entirely in your head in under 5 seconds. ### Common Pitfall A frequent mistake is misidentifying the cube root of decimals. Many students hastily assume the cube root of $0.027$ is $0.03$ instead of $0.3$. Always double-check decimal places: $0.3 \times 0.3 \times 0.3$ yields three decimal places ($0.027$). ### Final Answer **Therefore, the correct answer is 0.8.**