Simplify each expression. \( \left(2 a^{4}\right)^{-3}, \boldsymbol{a} \neq 0 \) \( \frac{1}{8 a^{7}} \) \( \frac{1}{2 a^{12}} \) \( \frac{1}{8 a^{12}} \) \( \frac{1}{2 a^{7}} \)
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Let's simplify each expression step by step! 1. To simplify \( \left(2 a^{4}\right)^{-3} \), we apply the negative exponent rule, which states that \( x^{-n} = \frac{1}{x^{n}} \). Thus: \[ \left(2 a^{4}\right)^{-3} = \frac{1}{(2 a^{4})^{3}} = \frac{1}{2^{3} (a^{4})^{3}} = \frac{1}{8 a^{12}}. \] 2. The expression \( \frac{1}{8 a^{7}} \) is already in its simplest form. Nothing further can be simplified here! 3. For \( \frac{1}{2 a^{12}} \), this is also in simplest form, straightforward and neat! 4. Again, \( \frac{1}{8 a^{12}} \) does not require any further simplification—it’s good as is! 5. Lastly, \( \frac{1}{2 a^{7}} \) is likewise in its simplest form and doesn't need any changes. So, in summary: - \( \left(2 a^{4}\right)^{-3} = \frac{1}{8 a^{12}} \) - No further simplifications for any others: - \( \frac{1}{8 a^{7}} \) - \( \frac{1}{2 a^{12}} \) - \( \frac{1}{8 a^{12}} \) - \( \frac{1}{2 a^{7}} \)
