Please answer the following questions: Simplify (t^2+3t-18)/(t^2+2t-15)* (t^2-3t-10)/(t^2+8t+12) (a^3+7a^2+10a)/(a^2-9a-70)   9/(7t+9) - what  is the number for which the rational expression is...

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As with simplifying numerical fractions, the key is to factor numerator and denominator and use the multiplicative identity property to eliminate common factors.

(1) `(t^2+3t-18)/(t^2+2t-15)*(t^2-3t-10)/(t^2+8t+12)`

`=((t+6)(t-3))/((t+5)(t-3))*((t-5)(t+2))/((t+6)(t+2))`

`=((t+6))/((t+5))*((t-5))/((t+6))`

`=(t-5)/(t+5)`

This rational expression fails to exist when t=-5. The original rational expression fails to exist when t=-5,3,-6, or -2.

(2) `(a^3+7a^2+10a)/(a^2-9a-70)`

`=(a(a^2+7a+10))/((a-14)(a+5))`

`=(a(a+2)(a+5))/((a-14)(a+5))`

`=(a(a+2))/((a-14))`  which is fully simplified.

This rational expression fails to exist for a=14. The original expression fails to exist when a=14 or -5.

(3) `9/(7t+9)` fails to exist if the denominator is zero.

`7t+9=0==>t=-9/7`

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