Question

Solve.
(5√5​)⁻²ˣ⁺¹=1/5​⋅125ˣ⁻³
x = ___

Answer 1

To solve the given equation, rewrite the expressions with simplified forms, equate the two sides, take the logarithm of both sides, simplify further, and finally solve for x.

Step-by-step explanation:

To solve the given equation, we need to simplify both sides. Let's start with the right side of the equation:

1. 
   
3. 1/5 multiplied by 125 gives us 125/5, which simplifies to 25.
4. 
   
6. x raised to the power of -3 can be written as 1/x^3.

Now, let's simplify the left side of the equation:

1. 
   
3. 5√5 raised to the power of -2x+1 can be written as (1/5√5)^(2x+1).

Now, we can equate the simplified expressions:

(1/5√5)^(2x+1) = 25 * (1/x^3)

To solve for x, we can take the logarithm of both sides:

log((1/5√5)^(2x+1)) = log(25 * (1/x^3))

Using logarithm properties, we can bring down the exponent:

(2x+1) * log(1/5√5) = log(25) + log(1/x^3)

Simplifying further:

(2x+1) * log(1/5√5) = log(25) - 3 * log(x)

Now, we can solve for x by isolating it:

2x+1 = (log(25) - 3 * log(x)) / log(1/5√5)

Finally, we can solve this equation for x using algebraic techniques.