Certainly! Let's solve the equation step-by-step:
Given equation:
[tex]\[
\left(\frac{4}{5}\right)^{3x + 2} = \left(\frac{5}{4}\right)^{x - 2}
\][/tex]
Step 1: Recognize that [tex]\(\left(\frac{5}{4}\right)\)[/tex] is the reciprocal of [tex]\(\left(\frac{4}{5}\right)\)[/tex]. Using this property, we can rewrite the equation:
[tex]\[
\left(\frac{4}{5}\right)^{3x + 2} = \left(\left(\frac{4}{5}\right)^{-1}\right)^{x - 2}
\][/tex]
Simplify the right-hand side:
[tex]\[
\left(\frac{4}{5}\right)^{3x + 2} = \left(\frac{4}{5}\right)^{-(x - 2)}
\][/tex]
Step 2: Since the bases are the same, we can set the exponents equal to each other. This gives us the equation:
[tex]\[
3x + 2 = -(x - 2)
\][/tex]
Step 3: Simplify the equation:
[tex]\[
3x + 2 = -x + 2
\][/tex]
Step 4: Combine like terms. Add [tex]\(x\)[/tex] to both sides:
[tex]\[
3x + 2 + x = 2
\][/tex]
[tex]\[
4x + 2 = 2
\][/tex]
Step 5: Subtract 2 from both sides:
[tex]\[
4x = 0
\][/tex]
Step 6: Divide both sides by 4:
[tex]\[
x = 0
\][/tex]
The solution to the equation is:
[tex]\[
x = 0
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\left(\frac{4}{5}\right)^{3x+2}=\left(\frac{5}{4}\right)^{x-2}\)[/tex] is:
[tex]\[
x = 0
\][/tex]
