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Question

Answer

$$b^2-\dfrac{4}{b^2}-4b+4=\dfrac{b^4-4b^3+4b^2-4}{b^2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}b^2-\frac{4}{b^2}-4b+4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{b^4-4}{b^2}-4b+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{b^4-4b^3-4}{b^2}+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{b^4-4b^3+4b^2-4}{b^2}\end{aligned} $$
①

Subtract $ \dfrac{4}{b^2} $ from $ b^2 $ to get $ \dfrac{ \color{purple}{ b^4-4 } }{ b^2 }$.

Step 1: Write $ b^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

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

Subtract $4b$ from $ \dfrac{b^4-4}{b^2} $ to get $ \dfrac{ \color{purple}{ b^4-4b^3-4 } }{ b^2 }$.

Step 1: Write $ 4b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} \frac{b^4-4}{b^2} -4b & \xlongequal{\text{Step 1}} \frac{b^4-4}{b^2} - \frac{4b}{\color{red}{1}} = \frac{ b^4-4 }{ b^2 } - \frac{ 4b \cdot \color{blue}{ b^2 }}{ 1 \cdot \color{blue}{ b^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ b^4-4 } }{ b^2 } - \frac{ \color{purple}{ 4b^3 } }{ b^2 }=\frac{ \color{purple}{ b^4-4b^3-4 } }{ b^2 } \end{aligned} $$
③

Add $ \dfrac{b^4-4b^3-4}{b^2} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ b^4-4b^3+4b^2-4 } }{ b^2 }$.

Step 1: Write $ 4 $ 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 second fraction by $\color{blue}{b^2}$.

$$ \begin{aligned} \frac{b^4-4b^3-4}{b^2} +4 & \xlongequal{\text{Step 1}} \frac{b^4-4b^3-4}{b^2} + \frac{4}{\color{red}{1}} = \frac{ b^4-4b^3-4 }{ b^2 } + \frac{ 4 \cdot \color{blue}{ b^2 }}{ 1 \cdot \color{blue}{ b^2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ b^4-4b^3-4 } }{ b^2 } + \frac{ \color{purple}{ 4b^2 } }{ b^2 }=\frac{ \color{purple}{ b^4-4b^3+4b^2-4 } }{ b^2 } \end{aligned} $$
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