Step-by-step explanation:
Here are the steps to solve the equation (5√(5))^(-2x + 1) = 1/5 ∙ 125^(x-3):
Express all terms with a common base:
Rewrite 5√(5) as 5^(3/2) (since √(5) = 5^(1/2)).
Rewrite 125 as 5^3.
Rewrite 1/5 as 5^(-1).
The equation becomes:
(5^(3/2))^(-2x + 1) = 5^(-1) ∙ (5^3)^(x-3)
Apply the power of a power rule:
(a^m)^n = a^(m*n)
5^((3/2)*(-2x + 1)) = 5^(-1) ∙ 5^(3(x-3))
Simplify exponents:
5^(-3x + 3/2) = 5^(3x - 9 - 1)
5^(-3x + 3/2) = 5^(3x - 10)
Equate exponents:
Since the bases are the same, we can equate the exponents:
-3x + 3/2 = 3x - 10
Solve for x:
Add 3x to both sides: 3/2 = 6x - 10
Add 10 to both sides: 23/2 = 6x
Divide both sides by 6: x = 23/12
Therefore, the solution to the equation is x = 23/12.
