If `x^(-((p)/(q)))= ((1)/(x))^(k)`, then k= _____

Answer

To solve the equation \( x^{-\frac{p}{q}} = \left(\frac{1}{x}\right)^{k} \), we will follow these steps:

### Step 1: Rewrite the Right Side
The expression \(\left(\frac{1}{x}\right)^{k}\) can be rewritten using the property of exponents:
\[ \left(\frac{1}{x}\right)^{k} = x^{-k} \]
So, we can rewrite the equation as:
\[ x^{-\frac{p}{q}} = x^{-k} \]

### Step 2: Set the Exponents Equal
Since the bases are the same (both are \(x\)), we can set the exponents equal to each other:
\[ -\frac{p}{q} = -k \]

### Step 3: Remove the Negative Sign
Multiplying both sides of the equation by -1 gives:
\[ \frac{p}{q} = k \]

### Conclusion
Thus, the value of \(k\) is:
\[ k = \frac{p}{q} \]

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