The value of $$ \left( \frac{0.1 \times 0.1 \times 0.1 + 0.02 \times 0.02 \times 0.02}{0.2 \times 0.2 \times 0.2 + 0.04 \times 0.04 \times 0.04} \right) $$ is

### Concept & Formula This problem tests your understanding of proportional relationships and extracting common factors from exponents. It does not use the $a^3+b^3$ identity, but rather scaling factors. The governing rule of exponents is: $$ (kx)^n = k^n x^n $$ ### Step-by-Step Solution Let the numerator be represented by $N$: $$ N = (0.1)^3 + (0.02)^3 $$ Now, analyze the numbers in the denominator. Notice that they are exactly twice the corresponding numbers in the numerator: * $0.2 = 2 \times 0.1$ * $0.04 = 2 \times 0.02$ Rewrite the denominator using this relationship: $$ (2 \times 0.1)^3 + (2 \times 0.02)^3 $$ Apply the exponent rule to expand the terms: $$ [2^3 \times (0.1)^3] + [2^3 \times (0.02)^3] $$ Factor out the common term $2^3$ (which equals $8$): $$ 8 \times [ (0.1)^3 + (0.02)^3 ] $$ Notice that the expression inside the brackets is exactly the numerator $N$. Substitute $N$ back into the full fraction: $$ \frac{N}{8N} $$ The complex decimal terms cancel out entirely, leaving: $$ \frac{1}{8} $$ Convert the simple fraction to a decimal: $$ \frac{1}{8} = 0.125 $$ ### Exam Strategy & Shortcut Look vertically instead of horizontally. The bottom numbers ($0.2$ and $0.04$) are exactly $2$ times the top numbers ($0.1$ and $0.02$). Because all terms are being cubed, the denominator is scaled by $2^3$, which is $8$. The complex terms cancel out instantly, leaving you with $1/8$. $1/8 = 0.125$. ### Common Pitfall The most common trap is seeing that the bottom terms are twice the top terms and erroneously concluding the answer is $1/2$ (or $0.5$). You must remember that the scaling factor ($2$) is also being cubed along with the decimals. ### Final Answer **Therefore, the correct answer is 0.125.**