Tap the blue circles to see an explanation.
$$ \begin{aligned}(-5sqrt\cdot2+5sqrt\cdot2i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(-10qrst+5sqrt\cdot2i)^3 \xlongequal{ } \\[1 em] & \xlongequal{ }(-10qrst+10iqrst)^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1000i^3q^3r^3s^3t^3-3000i^2q^3r^3s^3t^3+3000iq^3r^3s^3t^3-1000q^3r^3s^3t^3\end{aligned} $$ | |
① | $$ 5 s q r t \cdot 2 = 10 q r s t $$ |
② | Find $ \left(-10qrst+10iqrst\right)^3 $ in two steps. S1: Swap two terms inside bracket S2: apply formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 10iqrst $ and $ B = 10qrst $. $$ \left(-10qrst+10iqrst\right)^3 \xlongequal{ S1 } \left(10iqrst-10qrst\right)^3 = \left( 10iqrst \right)^3-3 \cdot \left( 10iqrst \right)^2 \cdot 10qrst + 3 \cdot 10iqrst \cdot \left( 10qrst \right)^2-\left( 10qrst \right)^3 = 1000i^3q^3r^3s^3t^3-3000i^2q^3r^3s^3t^3+3000iq^3r^3s^3t^3-1000q^3r^3s^3t^3 $$ |