1. **State the problem:** Solve for $x$ in terms of $n$ given the equation: $$\frac{18 \times (\sqrt{27})^{4n + 6}}{6 \times 9^{2n + 8}} = 3^n$$ 2. **Rewrite the bases in terms of 3:** Note that $\sqrt{27} = 27^{1/2} = (3^3)^{1/2} = 3^{3/2}$ and $9 = 3^2$. 3. **Substitute these into the equation:** $$\frac{18 \times (3^{3/2})^{4n + 6}}{6 \times (3^2)^{2n + 8}} = 3^n$$ 4. **Simplify the exponents:** $$\frac{18 \times 3^{(3/2)(4n + 6)}}{6 \times 3^{2(2n + 8)}} = 3^n$$ 5. **Simplify the constants:** $$\frac{18}{6} = 3$$ So the equation becomes: $$3 \times \frac{3^{(3/2)(4n + 6)}}{3^{2(2n + 8)}} = 3^n$$ 6. **Use the property of exponents to combine the fraction:** $$3 \times 3^{(3/2)(4n + 6) - 2(2n + 8)} = 3^n$$ 7. **Simplify the exponent in the numerator:** Calculate each term: $$(3/2)(4n + 6) = 6n + 9$$ $$2(2n + 8) = 4n + 16$$ So the exponent difference is: $$6n + 9 - (4n + 16) = 2n - 7$$ 8. **Rewrite the equation:** $$3 \times 3^{2n - 7} = 3^n$$ 9. **Combine the left side:** $$3^{1 + 2n - 7} = 3^n$$ $$3^{2n - 6} = 3^n$$ 10. **Set the exponents equal since bases are the same:** $$2n - 6 = n$$ 11. **Solve for $n$:** $$2n - n = 6$$ $$n = 6$$ 12. **Interpretation:** The equation holds true when $n = 6$. Since the problem asks to express $x$ in terms of $n$, but $x$ is not explicitly given in the original equation, it seems there might be a misunderstanding. If $x$ is intended to be the expression on the left side, then the equation relates $x$ and $n$ as: $$x = 3^n$$ Thus, $x$ is expressed in terms of $n$ as: $$\boxed{x = 3^n}$$ --- **Summary:** - We rewrote all terms with base 3. - Simplified exponents and constants. - Equated exponents to solve for $n$. - Expressed $x$ in terms of $n$ as $x = 3^n$.