Simplify each expression. Write your answer using a negative exponent. \( \begin{array}{ll}\text { i) } 7^{-12} \cdot 7^{5}=\square & \text { iii) } \frac{3^{4}}{3^{6}}=\square \\ \text { ii) } 10^{-10} \cdot 10^{5}=\square\end{array} \)

Let's simplify each expression step by step! i) For \( 7^{-12} \cdot 7^{5} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \). Thus: \[ 7^{-12} \cdot 7^{5} = 7^{-12 + 5} = 7^{-7} \] ii) Similarly, for \( 10^{-10} \cdot 10^{5} \): \[ 10^{-10} \cdot 10^{5} = 10^{-10 + 5} = 10^{-5} \] iii) For \( \frac{3^{4}}{3^{6}} \), we use the exponent rule for division, which is \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{3^{4}}{3^{6}} = 3^{4 - 6} = 3^{-2} \] So the answers are: i) \( 7^{-7} \), ii) \( 10^{-5} \), iii) \( 3^{-2} \).