Sure! Let's solve the given equation step by step. We need to solve the equation:
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \left(\frac{27}{125}\right)^{-1} \][/tex]
### Step 1: Simplify the right side of the equation
First, we can simplify the right-hand side of the equation. Recall that:
[tex]\[ \left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \][/tex]
So:
[tex]\[ \left(\frac{27}{125}\right)^{-1} = \frac{125}{27} \][/tex]
### Step 2: Substitute and rewrite the equation
Now the equation becomes:
[tex]\[ \left(\sqrt{\frac{3}{5}}\right)^{x-1} = \frac{125}{27} \][/tex]
### Step 3: Simplify the left side of the equation
Next, look at the left-hand side of the equation. Recall that:
[tex]\[ \sqrt{\frac{3}{5}} = \left(\frac{3}{5}\right)^{1/2} \][/tex]
So we rewrite the left-hand side as:
[tex]\[ \left(\left(\frac{3}{5}\right)^{1/2}\right)^{x-1} \][/tex]
Using the power of a power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we get:
[tex]\[ \left(\frac{3}{5}\right)^{(1/2)(x-1)} = \left(\frac{3}{5}\right)^{\frac{x-1}{2}} \][/tex]
### Step 4: Equate and solve exponents
We now have:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \frac{125}{27} \][/tex]
Notice that [tex]\(\frac{125}{27}\)[/tex] can be written as [tex]\(\left(\frac{5}{3}\right)^3\)[/tex]:
[tex]\[ \frac{125}{27} = \left(\frac{5}{3}\right)^3 \][/tex]
We then rewrite the equation as:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{5}{3}\right)^3 \][/tex]
### Step 5: Align bases and solve for x
Since [tex]\(\frac{3}{5}\)[/tex] is the reciprocal of [tex]\(\frac{5}{3}\)[/tex], we can rewrite [tex]\(\left(\frac{5}{3}\right)^3\)[/tex] as [tex]\(\left(\frac{3}{5}\right)^{-3}\)[/tex]:
[tex]\[ \left(\frac{3}{5}\right)^{\frac{x-1}{2}} = \left(\frac{3}{5}\right)^{-3} \][/tex]
If the bases are equal, we can set their exponents equal to each other:
[tex]\[ \frac{x-1}{2} = -3 \][/tex]
### Step 6: Solve for x
Multiply both sides by 2 to clear the fraction:
[tex]\[ x - 1 = -6 \][/tex]
Finally, add 1 to both sides to isolate x:
[tex]\[ x = -6 + 1 \][/tex]
[tex]\[ x = -5 \][/tex]
So, the solution to the equation is [tex]\( x = -5 \)[/tex].
