a+b+c = 4
a²+b²+c²=10
a³+b³+c³=22
a⁴+b⁴+c⁴=
what is the answer please tell thise who will solve right i will give thans and marks as brainlist answer
Answers
Answer:
a⁴+b⁴+c⁴ = 50
Step-by-step explanation:
a + b + c = 4_________(1)
a²+b²+c²=10________(2)
a³+b³+c³=22________(3)
Taking eq. 1
a + b + c = 4
Squaring both sides
(a+b+c)² = 4²
a²+b²+c²+2ab+2bc+2ca = 16
a²+b²+c²+2(ab+bc+ca) = 16
From eq. 2, putting value of a²+b²+c²
10 + 2(ab+bc+ca) = 16
2(ab+bc+ca) = 16 - 10
2(ab+bc+ca) = 6
(ab+bc+ca) = 6/2
(ab+bc+ca) = 3______________(4)
Multiplying eq. 1 and 2
a+b+c = 4
a²+b²+c² = 10
a³+ab²+ac²+ba²+b³+bc²+ca²+cb²+c³ = 40
a³+b³+c³+ab²+ac²+ba²+bc²+ca²+cb² = 40
Putting value of a³+b³+c³ given in eq. 3
= 22+ab²+ac²+ba²+bc²+ca²+cb² = 40
= ab²+ac²+ba²+bc²+ca²+cb² = 40-22
= ab²+ac²+ba²+bc²+ca²+cb² = 18____(5)
Taking eq. 2,
a²+b²+c² = 10
Squaring both sides
(a²+b²+c²)² = 10²
= a⁴+b⁴+c⁴+2a²b²+2b²c²+2c²a² = 100
= a⁴+b⁴+c⁴+2(a²b²+b²c²+c²a²) = 100____(6)
Multiplying eq. 1 and 4
a+b+c = 4
(ab+bc+ca) = 3
= a²b+abc+a²c+ab²+b²c+abc+abc+bc²+c²a = 12
= a²b+a²c+ab²+b²c+bc²+c²a+3abc = 12
Putting value of ab²+ac²+ba²+bc²+ca²+cb² given in eq. 5
= 18 + 3abc = 12
3abc = 12-18
3abc = -6
abc = -6/3
abc = -2
Squaring eq. 4
= (ab+bc+ca) = 3
= (ab+bc+ca)² = 3²
= a²b²+b²c²+c²a²+2ab²c+2bc²a+2cba² = 9
= a²b²+b²c²+c²a²+2(ab²c+abc²+a²bc) = 9
= a²b²+b²c²+c²a²+2×abc(b+c+a) = 9
Putting value of abc and a+b+c from eq.
= a²b²+b²c²+c²a²+2×-2(4) = 9
= a²b²+b²c²+c²a²-16 = 9
= a²b²+b²c²+c²a² = 9+16
= a²b²+b²c²+c²a² = 25______(7)
Putting value of eq. 7 in eq. 6
= a⁴+b⁴+c⁴+2(a²b²+b²c²+c²a²) = 100
= a⁴+b⁴+c⁴+2(25) = 100
= a⁴+b⁴+c⁴+50 = 100
= a⁴+b⁴+c⁴ = 100-50
= a⁴+b⁴+c⁴ = 50