Evaluate for x `(sqrt((5)/(3)))^(x-8)=((27)/(125))^(2x-3)`

Answer

To solve the equation \((\sqrt{\frac{5}{3}})^{x-8} = \left(\frac{27}{125}\right)^{2x-3}\), we will follow these steps:

### Step 1: Rewrite the square root and the fractions in terms of exponents
The square root can be expressed as an exponent of \( \frac{1}{2} \):
\[ \sqrt{\frac{5}{3}} = \left(\frac{5}{3}\right)^{\frac{1}{2}} \]
Thus, we can rewrite the left side of the equation:
\[ \left(\frac{5}{3}\right)^{\frac{1}{2}}^{x-8} = \left(\frac{5}{3}\right)^{\frac{x-8}{2}} \]

### Step 2: Rewrite the right side
Next, we simplify the right side:
\[ \frac{27}{125} = \frac{3^3}{5^3} = \left(\frac{3}{5}\right)^3 \]
So, we can express the right side as:
\[ \left(\frac{27}{125}\right)^{2x-3} = \left(\left(\frac{3}{5}\right)^3\right)^{2x-3} = \left(\frac{3}{5}\right)^{3(2x-3)} = \left(\frac{3}{5}\right)^{6x-9} \]

### Step 3: Set the bases equal
Now, we have:
\[ \left(\frac{5}{3}\right)^{\frac{x-8}{2}} = \left(\frac{3}{5}\right)^{6x-9} \]
We can rewrite \(\left(\frac{3}{5}\right)^{6x-9}\) as:
\[ \left(\frac{5}{3}\right)^{-(6x-9)} \]
Thus, we have:
\[ \left(\frac{5}{3}\right)^{\frac{x-8}{2}} = \left(\frac{5}{3}\right)^{-(6x-9)} \]

### Step 4: Set the exponents equal
Since the bases are equal, we can set the exponents equal to each other:
\[ \frac{x-8}{2} = -(6x-9) \]

### Step 5: Solve for \(x\)
Now, we solve the equation:
1. Multiply both sides by 2 to eliminate the fraction:
\[ x - 8 = -2(6x - 9) \]
\[ x - 8 = -12x + 18 \]
2. Rearranging gives:
\[ x + 12x = 18 + 8 \]
\[ 13x = 26 \]
3. Finally, divide by 13:
\[ x = 2 \]

### Final Answer
The value of \(x\) is \(2\).

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