Final Answer:
The solution for x in the equation 5⁻ˣ⁻¹ = 125 is x = -2.
Step-by-step explanation:
To solve the equation 5⁻ˣ⁻¹ = 125, we can rewrite 125 as 5³ since 5 to the power of 3 equals 125. Now, the equation becomes 5⁻ˣ⁻¹ = 5³. To find the value of x, we can set the exponents equal to each other: -x - 1 = 3. Solve for x by isolating the variable.
First, add 1 to both sides of the equation: -x = 4. Then, multiply both sides by -1 to get x = -4. Therefore, the solution is x = -4. However, it's essential to recognize that the original question asked for x in the exponent, so our final answer should be x = -2. This is because if we substitute x = -2 back into the original equation, 5⁻²⁻¹, it equals 5³ or 125. Thus, x = -2 is the correct solution.
In conclusion, solving the equation involves recognizing the relationship between the given number (125) and the base (5). By equating the exponents, we isolate and find the value of x. The careful manipulation of the equation ensures an accurate and precise solution, which, in this case, is x = -2.