Step 1: Understanding the Concept:
The given expression consists of terms in the form of \( \frac{a^3 - b^3}{a - b} \) and \( \frac{a^3 + b^3}{a + b} \) where \( a = \sqrt{5} \) and \( b = \sqrt{2} \).
We can use standard algebraic identities to simplify these fractions. : Key Formula or Approach:
1. \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \implies \frac{a^3 - b^3}{a - b} = a^2 + ab + b^2 \).
2. \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \implies \frac{a^3 + b^3}{a + b} = a^2 - ab + b^2 \). Step 2: Detailed Explanation:
Let \( a = \sqrt{5} \) and \( b = \sqrt{2} \).
Note that \( a^2 = 5 \), \( b^2 = 2 \), and \( ab = \sqrt{10} \).
The given expression is:
\[ E = \frac{a^3 - b^3}{a - b} + \frac{a^3 + b^3}{a + b} \]
Applying the identities:
\[ E = (a^2 + ab + b^2) + (a^2 - ab + b^2) \]
Simplifying by combining like terms:
\[ E = a^2 + ab + b^2 + a^2 - ab + b^2 \]
\[ E = 2a^2 + 2b^2 \]
\[ E = 2(a^2 + b^2) \]
Substitute the values of \( a^2 \) and \( b^2 \):
\[ E = 2(5 + 2) \]
\[ E = 2(7) = 14 \). Step 3: Final Answer:
The value of the expression is 14.