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Answer
The simplified form of \( \left(k^{-2}\right)^{3} \) is \( k^{-6} \) or \( \frac{1}{k^{6}} \).
Solution
1. We start with the expression:
\[
\left(k^{-2}\right)^{3}
\]
2. According to the exponent rule \(\left(a^m\right)^n = a^{mn}\), we multiply the exponents:
\[
\left(k^{-2}\right)^{3} = k^{-2 \cdot 3}
\]
3. Calculate the multiplication:
\[
-2 \cdot 3 = -6
\]
Therefore:
\[
k^{-2 \cdot 3} = k^{-6}
\]
4. The final simplified expression is:
\[
k^{-6}
\]
5. Alternatively, you can express the result with a positive exponent by recalling that \( k^{-6} = \frac{1}{k^{6}} \).
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Simplify this solution The Deep Dive
To simplify \( \left(k^{-2}\right)^{3} \), you apply the power of a power rule in exponents, which states that \( (a^{m})^{n} = a^{m \cdot n} \). So, you multiply the exponents: \[ (k^{-2})^{3} = k^{-2 \cdot 3} = k^{-6}. \] You can also express this in terms of positive exponents, giving you: \[ k^{-6} = \frac{1}{k^{6}}. \] Voilà! You've simplified it down to \( k^{-6} \) or \( \frac{1}{k^{6}} \).
