If `(sqrtx)^(y)=5`, what is the value of `(1)/(x^(2y))`?

Answer

To solve the equation \((\sqrt{x})^{y} = 5\) and find the value of \(\frac{1}{x^{2y}}\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (\sqrt{x})^{y} = 5 \]
This can be rewritten as:
\[ x^{\frac{y}{2}} = 5 \]

### Step 2: Square both sides
Next, we square both sides of the equation to eliminate the square root:
\[ (x^{\frac{y}{2}})^{2} = 5^{2} \]
This simplifies to:
\[ x^{y} = 25 \]

### Step 3: Find \(x^{2y}\)
Now, we need to find \(x^{2y}\). We can do this by squaring both sides of the equation \(x^{y} = 25\):
\[ (x^{y})^{2} = 25^{2} \]
This simplifies to:
\[ x^{2y} = 625 \]

### Step 4: Find \(\frac{1}{x^{2y}}\)
Now that we have \(x^{2y} = 625\), we can find \(\frac{1}{x^{2y}}\):
\[ \frac{1}{x^{2y}} = \frac{1}{625} \]

### Step 5: Simplify the fraction
To express \(\frac{1}{625}\) in decimal form:
\[ \frac{1}{625} = 0.0016 \]

### Final Answer
Thus, the value of \(\frac{1}{x^{2y}}\) is:
\[ \frac{1}{x^{2y}} = 0.0016 \]

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