Question 1
[tex]9^{2x+1}\text{ = }\frac{81^{x-2}}{3^x}[/tex]Step 1:
[tex]\begin{gathered} \text{Express 9 as = 3}^2 \\ 81=3^4 \end{gathered}[/tex]Step 2:
[tex]\begin{gathered} (3^2)^{2x+1}\text{ = }\frac{3^{4(x-2)}}{3^x} \\ 3^{4x+2}\text{ = }\frac{3^{4x-8}}{3^x} \\ 3^{4x+2}=3^{4x-8-x} \\ \text{Next, equate both the exponent} \\ 4x\text{ + 2 = 3x - 8} \\ \text{Collect similar terms} \\ 4x\text{ - 3x = -8 - 2} \\ x\text{ = -10} \end{gathered}[/tex]Question 2
[tex]\begin{gathered} 4^{x+1}\text{ }-9(2^x)\text{ = -2} \\ 2^{2(x+1)\text{ }}-9(2^x)\text{ + 2 = 0} \\ 2^{2x}\text{ }\times2^2-9(2^x)\text{ + 2 = 0} \\ 4(2^x)^2-9(2^x)\text{ + 2 = 0} \\ \text{Let 2}^x\text{ = p} \\ 4p^2\text{ - 9p + 2 = 0} \end{gathered}[/tex]Next, solve the equation to find the values of p.
[tex]\begin{gathered} 4p^2\text{ - 9p + 2 = 0} \\ 4p^2\text{ -8p - p + 2 = 0} \\ 4p(p\text{ - 2) - 1(p - 2) = 0} \\ (p\text{ - 2)(4p - 1) = 0} \\ p\text{ - 2 = 0 or 4p - 1 = 0} \\ p\text{ = 2 or }\frac{1}{4} \end{gathered}[/tex]Next, find the values of x, from the values of p.
[tex]\begin{gathered} p=2^x \\ 2=2^x \\ x\text{ = 1} \\ or \\ p=2^x \\ \frac{1}{4}=2^x \\ \frac{1}{2^2}=2^x \\ 2^{-2}=2^x \\ x\text{ = -2} \end{gathered}[/tex]Final answer
x = 1 or x = -2
Question 3
[tex]x\cdot\text{ y = x + y + 3xy}[/tex]Let the identity element = e
Since the operation is commutative
x * e = x
[tex]\begin{gathered} x\cdot\text{ e = x + e + 3xe = x} \\ e\text{ + 3xe = x - x} \\ e(\text{ 1 + 3x ) = 0} \\ e\text{ = }\frac{0}{1\text{ + 3x}} \\ e\text{ = 0} \\ \text{Identity element e = 0} \end{gathered}[/tex]Next,
[tex]\begin{gathered} \text{Let inverse element be y}^{-1} \\ \text{Therefore} \\ x\cdot y^{-1}\text{ = e} \\ x+y^{-1}\text{ + 3}xy^{-1}\text{ = 0} \\ y^{-1}(1\text{ + 3x) = -x} \\ y^{-1}\text{ = }\frac{-x}{1\text{ + 3x}} \end{gathered}[/tex][tex]\text{Inverse = }\frac{-x}{1\text{ + 3x}}[/tex]