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`-1`Solution :

`alpha ` is a complex fifth roots of unity. <br> `therefore 1+ alpha +alpha^(2) + alpha^(3)+ alpha^(4) = 0` <br> Now, given that <br> `(1+ alpha + alpha^(3) + alpha^(3))^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` <br> `rArr (-alpha^(4)) ^(2005) = p + qalpha + ralpha^(2) + salpha^(3)` <br> `rArr - alpha^(8020) = p+ qalpha + ralpha^(2)+ salpha^(3)` <br> `rArr -(alpha^(5))^(1640) = p +qalpha+ = ralpha^(2) + salpha^(3)` <br> `rArr -1 = p + qalpha + ralpha^(2) + salpha^(3)` <br> `rArr p = - 1, q = r = s = 0` <br> `therefore p + q+ r+s = -1`Transcript

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00:00 - 00:59 | the Alpha is complex 5th root of unity and oneplus alpha + Alpha square plus b cube wholesale solar power 2005 is equal to 1 + email + are Alpha square plus and Alpha q where p q r and S are real number then you have to find the value of P + Q + 8 + now we are given that Alpha is a complex 5th root root of unity that Alpha is the power 5 will be one and Alpha in complex 5th root of unity then 1 + alpha + Alpha square plus B plus l pareshani power 400 and if we consider this value is given a 1 + alpha + Alpha square plus |

01:00 - 01:59 | and fork you forest power 2005 list where given a p + Q alpha + are Alpha square plus is alpha you now from here oneplus alpha + Alpha square plus Alpha Q will be minus Alpha power for so this is minus power 4 hole SPD Power 2005 + 12 + are Alpha square plus is alpha hair design -1 list the power something on this -1 we can take outside and this is alpha rays to the power this will be 800 is equal to P + Q alpha |

02:00 - 02:59 | + are Alpha square plus is alpha you now here know the value of alphabets to the power 5 letter write try to write this power and the multiple of 5 this is -1 and hair this is the power 5 here we divide the power bi 5 that is I have taken 5 common than this will be 104 this is equal to P + us + are Alpha square plus at Alpha Q now we know that alpha rays to the power 5 is one so hair this will be -1 hair one power 1504 is equal to P + sin alpha + are Alpha square plus |

03:00 - 03:59 | is alpha Q now this one is the power of one only and one into minus one will be -1 this is minus 1 is equal to 3 + you alpha + R square + Alpha Q now every compare both side that is on comparing on comparing type we get hair the constant value will see and hear the constant value minus will be - 1000 hair there is no Alpha square |

04:00 - 04:59 | root of Alka Alpha square can say this is zero so I will be zero and similarly there we do not have Alpha q q q is equal to zero now we have to find the value of P + Q + 8 + 8 so if we compute P + q love R + S then this is minus 1 + 20 + 0 + 0 which is equals to minus the required value for discussion and we're done thank you |