Solve for $n$, if $9^n\cdot9^n\cdot9^n\cdot9^n=81^4$.
To solve for $n$, I need to express both sides of the equation in terms of the same base. Since 9 and 81 are both powers of 3, I can rewrite them using exponent rules. On the left side, I have $9^n\cdot9^n\cdot9^n\cdot9^n=(9^n)^4=(3^2)^{n\cdot4}=3^{2n\cdot4}$. On the right side, I have $81^4=(3^4)^4=3^{4\cdot4}$. Now I have $3^{2n\cdot4}=3^{4\cdot4}$, which means that the exponents must be equal. So I can set up an equation: $2n\cdot4=4\cdot4$. Dividing both sides by 4, I get $2n=4$. Dividing both sides by 2, I get $n=2$. 