Answer:
x = 2n - 6
Step-by-step explanation:
Given:
[tex]\dfrac{18 \times \left(\sqrt{27}\right)^{4n+6}}{6 \times 9^{2n+8}}=3^x[/tex]
Express all numbers as the product of primes:
[tex]\dfrac{2 \times 3^2 \times \left(\sqrt{3^3}\right)^{4n+6}}{2 \times 3 \times (3^2)^{2n+8}}=3^x[/tex]
Cancel the common factor 2 × 3:
[tex]\dfrac{3 \times \left(\sqrt{3^3}\right)^{4n+6}}{(3^2)^{2n+8}}=3^x[/tex]
Apply the radical to exponent rule to [tex]\sqrt{3^3}[/tex].
[tex]\boxed{\begin{array}{c}\underline{\textsf{Radical to Exponent Rule}}\\\\\sqrt[n]{a^m}=a^{\frac{m}{n}}\end{array}}[/tex]
Therefore:
[tex]\dfrac{3 \times \left(3^{\frac32}\right)^{4n+6}}{ (3^2)^{2n+8}}=3^x[/tex]
Apply the Power of a Power exponent rule, which states that when an exponent is raised to another exponent, we multiply the exponents together:
[tex]\dfrac{3 \times 3^{6n+9}}{3^{4n+16}}=3^x[/tex]
Rewrite 3 as 3¹:
[tex]\dfrac{3^1 \times 3^{6n+9}}{3^{4n+16}}=3^x[/tex]
Apply the Product exponent rule, which states that when multiplying powers with the same base, their exponents can be added:
[tex]\dfrac{3^{1+6n+9}}{3^{4n+16}}=3^x\\\\\\\\\dfrac{3^{6n+10}}{3^{4n+16}}=3^x[/tex]
Apply the Quotient exponent rule, which states that the division of two powers with the same base can be simplified by subtracting the exponent of the divisor from the exponent of the dividend:
[tex]3^{6n+10-(4n+16)}=3^x\\\\\\3^{6n+10-4n-16}=3^x\\\\\\3^{2n-6}=3^x\\\\\\[/tex]
If two exponential functions with the same base are equal, then their exponents must also be equal. Therefore:
[tex]2n-6=x[/tex]
So, x expressed in terms of n is:
[tex]\Large\boxed{\boxed{x = 2n - 6}}[/tex]
