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If xx is an integer such that (−5)6x=510+x(-5)^{6x} = 5^{10 + x}, what is the value of xx?

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Composition of linear and quadratic functions: find a valueright chevron icon

Full solution

Q. If xx is an integer such that (−5)6x=510+x(-5)^{6x} = 5^{10 + x}, what is the value of xx?
  1. Rewrite Bases: We need to make the bases the same, so let's rewrite (−5)6x(-5)^{6x} as 56x5^{6x}, but remember that the negative sign will add a factor of (−1)6x(-1)^{6x}.
  2. Compare Exponents: Since (−1)6x(-1)^{6x} is always 11 (because any even power of −1-1 is 11), we can ignore it and just compare the exponents of 56x5^{6x} and 510+x5^{10 + x}.
  3. Set Exponents Equal: Now we have 56x=510+x5^{6x} = 5^{10 + x}, so the exponents must be equal: 6x=10+x6x = 10 + x.
  4. Solve for x: Subtract xx from both sides to solve for xx: 6x−x=106x - x = 10.
  5. Divide to Find x: This simplifies to 5x=105x = 10.
  6. Final Answer: Divide both sides by 55 to find xx: x=105x = \frac{10}{5}.
  7. Final Answer: Divide both sides by 55 to find xx: x=105x = \frac{10}{5}.So, x=2x = 2.

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