Let's solve the equation step-by-step.
Given equation:
[tex]\[
\left[\left\{\left(\frac{2}{5}\right)^2\right\}^4\right]^{x+2}=\left[\left\{\left(\frac{2}{5}\right)^{-2}\right\}^{(x-1)}\right]^{-3}
\][/tex]
First, let's simplify both sides of the equation.
### Simplifying the Left Side
[tex]\[
\left\{\left(\frac{2}{5}\right)^2\right\}^4 = \left(\frac{2}{5}\right)^{2 \cdot 4} = \left(\frac{2}{5}\right)^8
\][/tex]
Now we have:
[tex]\[
\left(\left(\frac{2}{5}\right)^8\right)^{x+2} = \left(\frac{2}{5}\right)^{8(x+2)}
\][/tex]
### Simplifying the Right Side
[tex]\[
\left\{\left(\frac{2}{5}\right)^{-2}\right\}^{(x-1)} = \left(\frac{2}{5}\right)^{-2(x-1)}
\][/tex]
Raising this expression to the power of [tex]\(-3\)[/tex]:
[tex]\[
\left(\left(\frac{2}{5}\right)^{-2(x-1)}\right)^{-3} = \left(\frac{2}{5}\right)^{-2(x-1) \cdot (-3)}
\][/tex]
Simplifying the exponent:
[tex]\[
\left(\frac{2}{5}\right)^{-2(x-1) \cdot (-3)} = \left(\frac{2}{5}\right)^{6(x-1)}
\][/tex]
### Equating the Simplified Expressions
Now, equate the simplified expressions from both sides:
[tex]\[
\left(\frac{2}{5}\right)^{8(x+2)} = \left(\frac{2}{5}\right)^{6(x-1)}
\][/tex]
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
8(x+2) = 6(x-1)
\][/tex]
Solving the equation for [tex]\( x \)[/tex]:
1. Distribute the constants:
[tex]\[
8x + 16 = 6x - 6
\][/tex]
2. Collect the [tex]\( x \)[/tex]-terms on one side:
[tex]\[
8x - 6x = -6 - 16
\][/tex]
3. Simplify:
[tex]\[
2x = -22
\][/tex]
[tex]\[
x = -11
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = -2.75
\][/tex]
This concludes the solution to the equation.
