Question 17 of 20intermediate🔧 ApplyNumerical4 marks

Find the value of (a2a^{2} + \sqrt{a^{2} - 1})^4 + (a2a^{2} - \sqrt{a^{2} - 1})^4.

Correct Answer

2a82a^{8} + 12a612a^{6} - 10a410a^{4} - 4a24a^{2} + 2

Exercise: Miscellaneous Exercise on Chapter 7 | Q: 3 | (Page 8)
For More Understanding

Explanation

This problem requires applying the Binomial Theorem to expand both terms. The key insight is that when adding (x+y)^4 and (x-y)^4, the odd-powered terms cancel out, leaving only even-powered terms. This simplifies the calculation significantly. The textbook provides the binomial expansion formula and similar problems (like Question 11 and 12) that demonstrate this pattern of cancellation.

Solution Steps

  1. Step 1: Apply Binomial Theorem to find general formula for (x+y)^4 + (x-y)^4

  2. Using (a+b)^n = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙbⁿ, we get:

  3. (x+y)^4 = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

  4. (x-y)^4 = x⁴ - 4x³y + 6x²y² - 4xy³ + y⁴

  5. Step 2: Add the two expansions

  6. (x+y)^4 + (x-y)^4 = 2x⁴ + 12x²y² + 2y⁴ = 2(x⁴ + 6x²y² + y⁴)

  7. Step 3: Substitute x = a² and y = a2−1\sqrt{a²-1}

  8. Note: y² = a² - 1 and y⁴ = (a²-1)² = a⁴ - 2a² + 1

  9. Step 4: Simplify the expression

  10. = 2[(a²)⁴ + 6(a²)²(a²-1) + (a²-1)²]

  11. = 2[a⁸ + 6a⁴(a²-1) + a⁴ - 2a² + 1]

  12. = 2[a⁸ + 6a⁶ - 6a⁴ + a⁴ - 2a² + 1]

  13. = 2[a⁸ + 6a⁶ - 5a⁴ - 2a² + 1]

  14. Final Answer: 2a⁸ + 12a⁶ - 10a⁴ - 4a² + 2

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