Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{2pin(2pii\frac{n}{t}+b)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{2pin(2i^2p\frac{n}{t}+b)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{2pin(\frac{2i^2np}{t}+b)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{2pin\frac{2i^2np+bt}{t}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1}{\frac{4i^3n^2p^2+2binpt}{t}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{t}{4i^3n^2p^2+2binpt}\end{aligned} $$ | |
| ① | $$ 2 p i i = 2 i^{1 + 1} p = 2 i^2 p $$ |
| ② | Multiply $2i^2p$ by $ \dfrac{n}{t} $ to get $ \dfrac{ 2i^2np }{ t } $. Step 1: Write $ 2i^2p $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 2i^2p \cdot \frac{n}{t} & \xlongequal{\text{Step 1}} \frac{2i^2p}{\color{red}{1}} \cdot \frac{n}{t} \xlongequal{\text{Step 2}} \frac{ 2i^2p \cdot n }{ 1 \cdot t } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 2i^2np }{ t } \end{aligned} $$ |
| ③ | Add $ \dfrac{2i^2np}{t} $ and $ b $ to get $ \dfrac{ \color{purple}{ 2i^2np+bt } }{ t }$. Step 1: Write $ b $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ④ | Multiply $2inp$ by $ \dfrac{2i^2np+bt}{t} $ to get $ \dfrac{ 4i^3n^2p^2+2binpt }{ t } $. Step 1: Write $ 2inp $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 2inp \cdot \frac{2i^2np+bt}{t} & \xlongequal{\text{Step 1}} \frac{2inp}{\color{red}{1}} \cdot \frac{2i^2np+bt}{t} \xlongequal{\text{Step 2}} \frac{ 2inp \cdot \left( 2i^2np+bt \right) }{ 1 \cdot t } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4i^3n^2p^2+2binpt }{ t } \end{aligned} $$ |
| ⑤ | Divide $1$ by $ \dfrac{4i^3n^2p^2+2binpt}{t} $ to get $ \dfrac{ t }{ 4i^3n^2p^2+2binpt } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{ \frac{\color{blue}{4i^3n^2p^2+2binpt}}{\color{blue}{t}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{t}}{\color{blue}{4i^3n^2p^2+2binpt}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{t}{4i^3n^2p^2+2binpt} \xlongequal{\text{Step 3}} \frac{ 1 \cdot t }{ 1 \cdot \left( 4i^3n^2p^2+2binpt \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ t }{ 4i^3n^2p^2+2binpt } \end{aligned} $$ |