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Answer:
We conclude that:
[tex]\left(9c^{-9}\right)^{-3}=\frac{c^{27}}{729}[/tex]
Step-by-step explanation:
Given the expression
[tex]\left(9c^{-9}\right)^{-3}[/tex]
Apply exponent rule: [tex]a^{-b}=\frac{1}{a^b}[/tex]
[tex]\left(9c^{-9}\right)^{-3}=\frac{1}{\left(9c^{-9}\right)^3}[/tex]
Let us first solve:
[tex]\left(9c^{-9}\right)^3[/tex]
Apply exponent rule: [tex]\left(a\cdot \:b\right)^n=a^nb^n[/tex]
[tex]\left(9c^{-9}\right)^3=9^3\left(c^{-9}\right)^3[/tex]
[tex]=729\left(c^{-9}\right)^3[/tex]
Apply exponent rule: [tex]\left(a^b\right)^c=a^{bc},\:\quad \mathrm{\:assuming\:}a\ge 0[/tex]
[tex]=729c^{-9\cdot \:3}[/tex]
[tex]=729c^{-27}[/tex]
Apply exponent rule: [tex]a^{-b}=\frac{1}{a^b}[/tex]
[tex]=729\cdot \frac{1}{c^{27}}[/tex]
[tex]=\frac{729}{c^{27}}[/tex]
Therefore, the expression [tex]\left(9c^{-9}\right)^{-3}=\frac{1}{\left(9c^{-9}\right)^3}[/tex] becomes
[tex]\left(9c^{-9}\right)^{-3}=\frac{1}{\left(9c^{-9}\right)^3}[/tex]
[tex]=\frac{1}{\frac{729}{c^{27}}}[/tex] ∵ [tex]\left(9c^{-9}\right)^3=\frac{729}{c^{27}}[/tex]
[tex]=\frac{c^{27}}{729}[/tex] ∵ [tex]\frac{1}{\frac{b}{c}}=\frac{c}{b}[/tex]
Hence, we conclude that:
[tex]\left(9c^{-9}\right)^{-3}=\frac{c^{27}}{729}[/tex]