$$ \frac{(0.6)^4 - (0.5)^4}{(0.6)^2 + (0.5)^2} $$ is equal to

### Concept & Formula This expression uses a higher-power difference of squares. Any expression in the form of $a^4 - b^4$ can be factored as a difference of squares by treating it as $(a^2)^2 - (b^2)^2$. $$ x^2 - y^2 = (x - y)(x + y) \implies a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) $$ ### Step-by-Step Solution * Express the numerator using the identity framework where $a = 0.6$ and $b = 0.5$: $$ (0.6)^4 - (0.5)^4 = ((0.6)^2 - (0.5)^2)((0.6)^2 + (0.5)^2) $$ * Substitute this expansion back into the original fraction format: $$ \frac{((0.6)^2 - (0.5)^2)((0.6)^2 + (0.5)^2)}{(0.6)^2 + (0.5)^2} $$ * Cancel the common factor $((0.6)^2 + (0.5)^2)$ from both the numerator and the denominator. * This leaves you with: $$ (0.6)^2 - (0.5)^2 $$ * Compute the squares and find the difference: $$ 0.36 - 0.25 = 0.11 $$ ### Exam Strategy & Shortcut Recognize the identity form $\frac{a^4 - b^4}{a^2 + b^2} = a^2 - b^2$ automatically. Skip any intermediate writing steps and jump directly to computing $(0.6)^2 - (0.5)^2 = 0.36 - 0.25 = 0.11$. ### Common Pitfall Selecting option (a) $0.1$ by mistakenly computing $0.6 - 0.5$ instead of keeping the remaining terms squared ($(0.6)^2 - (0.5)^2$). ### Final Answer Therefore, the correct answer is 0.11.