To simplify the expression \(-9k^{-3} - 9k^{-4}\), follow these steps:
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Notice both terms have a common factor of \(-9\). Factor that out:
\[ -9 (k^{-3} + k^{-4}) \]
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Rewrite the expression inside the parentheses. Since \(k^{-4} = \frac{1}{k^4}\), we can express both terms with a common denominator:
\[ k^{-3} = \frac{1}{k^3} \] \[ k^{-4} = \frac{1}{k^4} \]
So, we need a common denominator, which will be \(k^4\):
\[ k^{-3} = \frac{k^1}{k^4} \quad \text{(since } \frac{1}{k^3} \cdot \frac{k}{k} = \frac{k}{k^4}\text{)} \]
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Rewrite the expression inside the parentheses using the common denominator:
\[ k^{-3} + k^{-4} = \frac{k}{k^4} + \frac{1}{k^4} = \frac{k + 1}{k^4} \]
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Combining everything back together, we have:
\[ -9(k^{-3} + k^{-4}) = -9 \left(\frac{k + 1}{k^4}\right) = -\frac{9(k + 1)}{k^4} \]
Thus, the simplified form of the original expression \(-9k^{-3} - 9k^{-4}\) is:
\[ -\frac{9(k + 1)}{k^4} \]