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Solve the exponential equation for x. 125^(-x-7)*5^(6x-12)=125^(2x+3) x=

Solve the exponential equation for x x .\newline125−x−7⋅56x−12=1252x+3 125^{-x-7} \cdot 5^{6 x-12}=125^{2 x+3} \newlinex= x=

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Q. Solve the exponential equation for x x .\newline125−x−7⋅56x−12=1252x+3 125^{-x-7} \cdot 5^{6 x-12}=125^{2 x+3} \newlinex= x=
  1. Express with Same Base: First, we need to express all terms with the same base to simplify the equation. We know that 125125 is 535^3. Let's rewrite the equation using this base.
  2. Rewrite Exponents: Rewrite 125(−x−7)125^{(-x-7)} as (53)(−x−7)(5^3)^{(-x-7)} and 125(2x+3)125^{(2x+3)} as (53)(2x+3)(5^3)^{(2x+3)}. The equation becomes (53)(−x−7)×5(6x−12)=(53)(2x+3)(5^3)^{(-x-7)} \times 5^{(6x-12)} = (5^3)^{(2x+3)}.
  3. Apply Power Rule: Apply the power of a power rule to simplify the exponents. This means multiplying the exponents inside the parentheses.\newlineThe equation becomes 53(−x−7)×56x−12=53(2x+3)5^{3(-x-7)} \times 5^{6x-12} = 5^{3(2x+3)}.
  4. Simplify Exponents: Simplify the exponents by performing the multiplications.\newlineThe equation becomes 5(−3x−21)×5(6x−12)=5(6x+9)5^{(-3x-21)} \times 5^{(6x-12)} = 5^{(6x+9)}.
  5. Set Exponents Equal: Since the bases are the same, we can set the exponents equal to each other. This gives us the equation −3x−21+6x−12=6x+9-3x - 21 + 6x - 12 = 6x + 9.
  6. Combine Like Terms: Combine like terms on the left side of the equation.\newlineThis gives us 3x−33=6x+93x - 33 = 6x + 9.
  7. Isolate x Term: Subtract 6x6x from both sides to get all x terms on one side.\newlineThis gives us 3x−6x−33=93x - 6x - 33 = 9.
  8. Simplify xx Terms: Simplify the xx terms by combining them.\newlineThis gives us −3x−33=9-3x - 33 = 9.
  9. Add to Isolate xx: Add 3333 to both sides to isolate the xx term.\newlineThis gives us −3x=9+33-3x = 9 + 33.
  10. Divide to Solve xx: Simplify the right side of the equation.\newlineThis gives us −3x=42-3x = 42.
  11. Calculate Value of x: Divide both sides by −3-3 to solve for xx. This gives us x=42−3x = \frac{42}{-3}.
  12. Calculate Value of x: Divide both sides by −3-3 to solve for x.\newlineThis gives us x=42−3x = \frac{42}{-3}.Calculate the division to find the value of x.\newlineThis gives us x=−14x = -14.

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