Answer
$$sqrt\cdot\frac{3}{3}(sqrt\cdot\frac{3}{3}+1)(sqrt\cdot\frac{3}{3}-1)=q^3r^3s^3t^3-qrst$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}sqrt\cdot\frac{3}{3}(sqrt\cdot\frac{3}{3}+1)(sqrt\cdot\frac{3}{3}-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\left(qrst \cdot \frac{ 3 : \color{orangered}{ 3 } }{ 3 : \color{orangered}{ 3 }} \cdot \left(qrst \cdot \frac{ 3 : \color{orangered}{ 3 } }{ 3 : \color{orangered}{ 3 }} + 1\right)\right) \cdot \left(qrst \cdot \frac{ 3 : \color{orangered}{ 3 } }{ 3 : \color{orangered}{ 3 }} - 1\right) \xlongequal{ } \\[1 em] & \xlongequal{ }sqrt\cdot\frac{1}{1}(sqrt\cdot\frac{1}{1}+1)(sqrt\cdot\frac{1}{1}-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}sqrt\cdot1(sqrt\cdot1+1)(sqrt\cdot1-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(1q^2r^2s^2t^2+qrst)(sqrt\cdot1-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}q^3r^3s^3t^3-q^2r^2s^2t^2+q^2r^2s^2t^2-qrst \xlongequal{ } \\[1 em] & \xlongequal{ }q^3r^3s^3t^3 -\cancel{q^2r^2s^2t^2}+ \cancel{q^2r^2s^2t^2}-qrst \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}q^3r^3s^3t^3-qrst\end{aligned} $$ | |
| ① | Divide both the top and bottom numbers by $ \color{orangered}{ 3 } $. |
| ② | Divide both the top and bottom numbers by $ \color{orangered}{ 3 } $. |
| ③ | Divide both the top and bottom numbers by $ \color{orangered}{ 3 } $. |
| ④ | Remove 1 from denominator. |
| ⑤ | Remove 1 from denominator. |
| ⑥ | Remove 1 from denominator. |
| ⑦ | Multiply $ \color{blue}{qrst} $ by $ \left( qrst+1\right) $ $$ \color{blue}{qrst} \cdot \left( qrst+1\right) = q^2r^2s^2t^2+qrst $$ |
| ⑧ | Multiply each term of $ \left( \color{blue}{q^2r^2s^2t^2+qrst}\right) $ by each term in $ \left( qrst-1\right) $. $$ \left( \color{blue}{q^2r^2s^2t^2+qrst}\right) \cdot \left( qrst-1\right) = \\ = q^3r^3s^3t^3 -\cancel{q^2r^2s^2t^2}+ \cancel{q^2r^2s^2t^2}-qrst $$ |
| ⑨ | Combine like terms: $$ q^3r^3s^3t^3 \, \color{blue}{ -\cancel{q^2r^2s^2t^2}} \,+ \, \color{blue}{ \cancel{q^2r^2s^2t^2}} \,-qrst = q^3r^3s^3t^3-qrst $$ |
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