multiply and simplify. \n\\( \\frac { t ^ { 3 } - 5 t } { t - t ^ { 4 } } \\cdot \\frac { t ^ { 4 } - t } {…

multiply and simplify. \n\\( \\frac { t ^ { 3 } - 5 t } { t - t ^ { 4 } } \\cdot \\frac { t ^ { 4 } - t } { 5 t - t ^ { 3 } } \\)\n\\( \\frac { t ^ { 3 } - 5 t } { t - t ^ { 4 } } \\cdot \\frac { t ^ { 4 } - t } { 5 t - t ^ { 3 } } = \\)

multiply and simplify. \n\\( \\frac { t ^ { 3 } - 5 t } { t - t ^ { 4 } } \\cdot \\frac { t ^ { 4 } - t } { 5 t - t ^ { 3 } } \\)\n\\( \\frac { t ^ { 3 } - 5 t } { t - t ^ { 4 } } \\cdot \\frac { t ^ { 4 } - t } { 5 t - t ^ { 3 } } = \\)

Answer

Explanation:

Step1: Factor numerators and denominators

  • Factor (t^{3}-5t=t(t^{2} - 5))
  • Factor (t^{4}-t=t(t^{3}-1)=t(t - 1)(t^{2}+t + 1))
  • Factor (t-t^{4}=t(1 - t^{3})=t(1 - t)(t^{2}+t + 1))
  • Factor (5t-t^{3}=t(5 - t^{2}))

So the expression becomes (\frac{t(t^{2}-5)}{t(1 - t)(t^{2}+t + 1)}\cdot\frac{t(t - 1)(t^{2}+t + 1)}{t(5 - t^{2})})

Step2: Simplify the expression

Notice that (t^{2}-5=-(5 - t^{2})) and (t - 1=-(1 - t))

[ \begin{align*} &\frac{t(t^{2}-5)}{t(1 - t)(t^{2}+t + 1)}\cdot\frac{t(t - 1)(t^{2}+t + 1)}{t(5 - t^{2})}\ =&\frac{t\cdot(-1)(5 - t^{2})}{t\cdot(-1)(t - 1)(t^{2}+t + 1)}\cdot\frac{t(t - 1)(t^{2}+t + 1)}{t(5 - t^{2})}\ =& 1 \end{align*} ]

Answer:

(1)