Question

This are the questions, I hope you able to get an answer for me

Answer 1

Question 1

`9^(2x+1)\text{ = }(81^(x-2))/(3^x)`

Step 1:

`\begin{gathered} \text{Express 9 as = 3}^2 \\ 81=3^4 \end{gathered}`

Step 2:

`\begin{gathered} (3^2)^(2x+1)\text{ = }(3^(4(x-2)))/(3^x) \\ 3^(4x+2)\text{ = }(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}`

Question 2

`\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{ }*2^2-9(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}`

Next, solve the equation to find the values of p.

`\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 }(1)/(4) \end{gathered}`

Next, find the values of x, from the values of p.

`\begin{gathered} p=2^x \\ 2=2^x \\ x\text{ = 1} \\ or \\ p=2^x \\ (1)/(4)=2^x \\ (1)/(2^2)=2^x \\ 2^(-2)=2^x \\ x\text{ = -2} \end{gathered}`

Final answer

x = 1 or x = -2

Question 3

`x\cdot\text{ y = x + y + 3xy}`

Let the identity element = e

Since the operation is commutative

x * e = x

`\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}`

Next,

`\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}`
`\text{Inverse = }\frac{-x}{1\text{ + 3x}}`