Find $ \left(9x+2\right)^3 $ using formula

$$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$

where $ A = 9x $ and $ B = 2 $.

$$ \left(9x+2\right)^3 = \left( 9x \right)^3+3 \cdot \left( 9x \right)^2 \cdot 2 + 3 \cdot 9x \cdot 2^2+2^3 = 729x^3+486x^2+108x+8 $$

Add $x$ and $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ x^2+1 } }{ x }$.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x }$.

$$ \begin{aligned} x+ \frac{1}{x} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{1}{x} = \frac{ x \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 1 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2 } }{ x } + \frac{ \color{purple}{ 1 } }{ x }=\frac{ \color{purple}{ x^2+1 } }{ x } \end{aligned} $$

Multiply $ \dfrac{x^2+1}{x} $ by $ 3x+9x^3 $ to get $ \dfrac{9x^5+12x^3+3x}{x} $.

Step 1: Write $ 3x+9x^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x^2+1}{x} \cdot 3x+9x^3 & \xlongequal{\text{Step 1}} \frac{x^2+1}{x} \cdot \frac{3x+9x^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^2+1 \right) \cdot \left( 3x+9x^3 \right) }{ x \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3x^3+9x^5+3x+9x^3 }{ x } = \frac{9x^5+12x^3+3x}{x} \end{aligned} $$

Multiply $ \dfrac{9x^5+12x^3+3x}{x} $ by $ 729x^3+486x^2+108x+8 $ to get $ \dfrac{6561x^8+4374x^7+9720x^6+5904x^5+3483x^4+1554x^3+324x^2+24x}{x} $.

Step 1: Write $ 729x^3+486x^2+108x+8 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{9x^5+12x^3+3x}{x} \cdot 729x^3+486x^2+108x+8 & \xlongequal{\text{Step 1}} \frac{9x^5+12x^3+3x}{x} \cdot \frac{729x^3+486x^2+108x+8}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 9x^5+12x^3+3x \right) \cdot \left( 729x^3+486x^2+108x+8 \right) }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 6561x^8+4374x^7+972x^6+72x^5+8748x^6+5832x^5+1296x^4+96x^3+2187x^4+1458x^3+324x^2+24x }{ x } = \\[1ex] &= \frac{6561x^8+4374x^7+9720x^6+5904x^5+3483x^4+1554x^3+324x^2+24x}{x} \end{aligned} $$