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The number of integer solutions of the equation \((x^2−10)^{(x2−3x−10)}=1\) is

Updated On: Jan 15, 2026
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Solution and Explanation

Given that any non-zero number raised to the power of 0 equals 1, and 1 raised to any power equals 1, the equation \( (x^2 - 10)^{(x^2 - 3x - 10)} = 1 \) holds true under the following conditions:

  • The base \( (x^2 - 10) \) is equal to 1.
  • The base \( (x^2 - 10) \) is equal to -1 (requires verification).
  • The exponent \( (x^2 - 3x - 10) \) is equal to 0.

Case 1: Base equals 1

If \( x^2 - 10 = 1 \), then \( x^2 = 11 \), which yields \( x = \pm \sqrt{11} \). Both \( x = \sqrt{11} \) and \( x = -\sqrt{11} \) are valid solutions as they result in the base being 1.

Case 2: Exponent equals 0

Setting the exponent to zero: \( x^2 - 3x - 10 = 0 \). Factoring gives \( (x - 5)(x + 2) = 0 \), leading to solutions \( x = 5 \) and \( x = -2 \).

Verification of these solutions in the original equation:

  • For \( x = 5 \): \( (5^2 - 10)^0 = (25 - 10)^0 = 15^0 = 1 \).
  • For \( x = -2 \): \( ((-2)^2 - 10)^0 = (4 - 10)^0 = (-6)^0 = 1 \).

Summary of Valid Solutions

  • \( x = \sqrt{11} \)
  • \( x = -\sqrt{11} \)
  • \( x = 5 \)
  • \( x = -2 \)

These four values satisfy the equation \( (x^2 - 10)^{(x^2 - 3x - 10)} = 1 \).

Final Answer

Total number of solutions: \( \boxed{4} \)

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