If a = 21 and b = 19, find the value of \(\frac{ a^{2} + b^{2} + ab }{ a^{3} - b^{3} } \)
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First, we need to compute each component of the expression \((a^2 + b^2 + ab) / (a^3 - b^3)\).
Given that \(a = 21\) and \(b = 19\), substitute these values into the expression.
Step 1: Calculate \(a^2\) and \(b^2\):
\(a^2 = 21^2 = 441\)
\(b^2 = 19^2 = 361\)
Step 2: Calculate \(ab\):
\(ab = 21 \times 19 = 399\)
Step 3: Calculate \(a^2 + b^2 + ab\):
\(a^2 + b^2 + ab = 441 + 361 + 399 = 1201\)
Step 4: Calculate \(a^3\) and \(b^3\):
\(a^3 = 21^3 = 9261\)
\(b^3 = 19^3 = 6859\)
Step 5: Calculate \(a^3 - b^3\):
\(a^3 - b^3 = 9261 - 6859 = 2402\)
Step 6: Substitute these results into the main expression:
\(\frac{a^2 + b^2 + ab}{a^3 - b^3} = \frac{1201}{2402}\)
Step 7: Simplify \(\frac{1201}{2402}\):
\(\frac{1201}{2402} = \frac{1201 \div 1201}{2402 \div 1201} = \frac{1}{2}\)
Thus, the value of \(\frac{a^2 + b^2 + ab}{a^3 - b^3}\) is \(\frac{1}{2}\).
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