If $(\sqrt{3})^5 \times 9^2 = 3^n \times 3\sqrt{3}$, then the value of $n$ is

### Concept & Formula The core concept is to convert all bases and surds (roots) into a common prime base, which in this case is $3$. By expressing all terms as powers of $3$, we can use the laws of indices to combine terms and equate the exponents on both sides of the equation. The necessary laws of indices are: $$\sqrt{x} = x^{\frac{1}{2}}$$ $$x^m \times x^n = x^{m+n}$$ $$(x^m)^n = x^{mn}$$ ### Step-by-Step Solution * **Given:** $$(\sqrt{3})^5 \times 9^2 = 3^n \times 3\sqrt{3}$$ * **Calculation:** Convert all terms to base $3$. For the left-hand side (LHS): $\sqrt{3} = 3^{\frac{1}{2}}$, so $(\sqrt{3})^5 = (3^{\frac{1}{2}})^5 = 3^{\frac{5}{2}} = 3^{2.5}$ $9 = 3^2$, so $9^2 = (3^2)^2 = 3^4$ Multiply these together using the product rule (add the exponents): LHS $= 3^{2.5} \times 3^4 = 3^{2.5 + 4} = 3^{6.5}$ For the right-hand side (RHS): $3\sqrt{3} = 3^1 \times 3^{\frac{1}{2}} = 3^{1 + 0.5} = 3^{1.5}$ Multiply this with $3^n$: RHS $= 3^n \times 3^{1.5} = 3^{n + 1.5}$ Equate the simplified LHS and RHS: $$3^{6.5} = 3^{n + 1.5}$$ Since the bases are identical, equate the exponents: $$6.5 = n + 1.5$$ $$n = 6.5 - 1.5$$ $$n = 5$$ ### Exam Strategy & Shortcut **Work Exclusively with Exponents:** Once you identify the common base is $3$, skip writing the base and only write the arithmetic for the exponents. LHS Exponents: $5 \times 0.5 + 2 \times 2 = 2.5 + 4 = 6.5$ RHS Exponents: $n + 1 + 0.5 = n + 1.5$ Set them equal: $6.5 = n + 1.5 \Rightarrow n = 5$. This can be done entirely mentally in 15 seconds. ### Common Pitfall Students often misinterpret $3\sqrt{3}$. They might accidentally treat it as an addition or miscalculate the combined power as just $3^{\frac{1}{2}}$ instead of recognizing it is $3^1 \times 3^{\frac{1}{2}} = 3^{\frac{3}{2}}$. ### Final Answer **Therefore, the correct answer is 5.**