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 $x^2$ and $ \dfrac{1}{x^2} $ to get $ \dfrac{ \color{purple}{ x^4+1 } }{ x^2 }$.

Step 1: Write $ x^2 $ 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 $\color{blue}{ x^2 }$.

$$ \begin{aligned} x^2+ \frac{1}{x^2} & \xlongequal{\text{Step 1}} \frac{x^2}{\color{red}{1}} + \frac{1}{x^2} = \frac{ x^2 \cdot \color{blue}{ x^2 }}{ 1 \cdot \color{blue}{ x^2 }} + \frac{ 1 }{ x^2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^4 } }{ x^2 } + \frac{ \color{purple}{ 1 } }{ x^2 }=\frac{ \color{purple}{ x^4+1 } }{ x^2 } \end{aligned} $$

Multiply $ \dfrac{x^4+1}{x^2} $ by $ 729x^3+486x^2+108x+8 $ to get $ \dfrac{ 729x^7+486x^6+108x^5+8x^4+729x^3+486x^2+108x+8 }{ x^2 } $.

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.

$$ \begin{aligned} \frac{x^4+1}{x^2} \cdot 729x^3+486x^2+108x+8 & \xlongequal{\text{Step 1}} \frac{x^4+1}{x^2} \cdot \frac{729x^3+486x^2+108x+8}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^4+1 \right) \cdot \left( 729x^3+486x^2+108x+8 \right) }{ x^2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 729x^7+486x^6+108x^5+8x^4+729x^3+486x^2+108x+8 }{ x^2 } \end{aligned} $$