Solve ` : {(625^(-(1)/(2))) ^(-(1)/(4))}=(0.2)^(4-3x)`

Answer

To solve the equation \( (625^{-\frac{1}{2}})^{-\frac{1}{4}} = (0.2)^{4 - 3x} \), we will follow these steps:

### Step 1: Simplify the Left-Hand Side (LHS)
We start with the LHS:
\[ (625^{-\frac{1}{2}})^{-\frac{1}{4}} \]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\), we can rewrite this as:
\[ 625^{(-\frac{1}{2}) \cdot (-\frac{1}{4})} = 625^{\frac{1}{8}} \]

### Step 2: Express 625 as a Power of 5
Next, we express 625 as a power of 5:
\[ 625 = 5^4 \]
Thus, we can substitute this into our expression:
\[ (5^4)^{\frac{1}{8}} = 5^{4 \cdot \frac{1}{8}} = 5^{\frac{4}{8}} = 5^{\frac{1}{2}} \]

### Step 3: Simplify the Right-Hand Side (RHS)
Now, we simplify the RHS:
\[ (0.2)^{4 - 3x} \]
First, we convert 0.2 into a fraction:
\[ 0.2 = \frac{2}{10} = \frac{1}{5} \]
Now we can express this as:
\[ \left(\frac{1}{5}\right)^{4 - 3x} = (5^{-1})^{4 - 3x} \]
Using the property of exponents again:
\[ 5^{-(4 - 3x)} = 5^{3x - 4} \]

### Step 4: Set the LHS Equal to the RHS
Now we have:
\[ 5^{\frac{1}{2}} = 5^{3x - 4} \]
Since the bases are the same, we can equate the exponents:
\[ \frac{1}{2} = 3x - 4 \]

### Step 5: Solve for x
To solve for \(x\), we first add 4 to both sides:
\[ \frac{1}{2} + 4 = 3x \]
Converting 4 to a fraction:
\[ \frac{1}{2} + \frac{8}{2} = 3x \implies \frac{9}{2} = 3x \]
Now, divide both sides by 3:
\[ x = \frac{9}{2} \cdot \frac{1}{3} = \frac{9}{6} = \frac{3}{2} \]

### Final Answer
Thus, the solution to the equation is:
\[ x = \frac{3}{2} \]