if a+b+c=1,a²+b²+c²=2,a³+b³+c³=3 then what's the value of a⁴+b⁴+c⁴
Answers
Answer:
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Answer:
The value of a⁵ + b⁵ + c⁵ is 6.
Step-by-step explanation:
Given : a + b + c = 1
a² + b² + c² = 2
a³ + b³ + c³ = 3
To Find : a⁵ + b⁵ + c⁵ = ?
Solution:
a² + b² + c² = 2
a³ + b³ + c³ = 3
Multiplying both
a⁵ + b⁵ + c⁵ + a²( b³ + c³) + b²(a³ + c³) + c²(a³ + b³ ) = 6
=> a⁵ + b⁵ + c⁵ + a²b² (a + b) +a²c²(a + c) + b²c²(b + c) = 6
=> a⁵ + b⁵ + c⁵ + a²b² (1-c) +a²c²(1-b) + b²c²(1-a) = 6
=> a⁵ + b⁵ + c⁵ + a²b² +a²c² + b²c² - abc(ab + ac + bc) = 6 Eq1
a + b + c = 1
squaring both sides
=> a² + b² + c² + 2(ab + ac + bc) = 1
=> 2 + 2(ab + ac + bc) = 1
=> ab + ac + bc = - 1/2
Squaring both sides
=> a²b² +a²c² + b²c² + 2abc(a + b + c) = 1/4
=> a²b² +a²c² + b²c² + 2abc = 1/4
=> a²b² +a²c² + b²c² = 1/4 - 2abc
a⁵ + b⁵ + c⁵ + a²b² +a²c² + b²c² - abc(ab + ac + bc) = 6
=> a⁵ + b⁵ + c⁵ + 1/4 - 2abc - abc (-1/2) = 6
=> a⁵ + b⁵ + c⁵ = 23/4 + 3abc/2
a + b + c = 1
a² + b² + c² = 2
Multiplying both
a³ + b³ + c³ + a(b² + c² ) + b(a² + c²) + c(a² + b²) = 2
=> 3 + ab (a + b) + ac(a + c) + bc(b + c) = 2
=> ab(1 - c) + ac( 1 - b) + bc( 1 - a) = - 1
=> ab + ac + bc -3abc = - 1
ab + ac + bc = - 1/2
=> - 1/2 - 3abc = - 1
=> 3abc = 1/2
=> abc = 1/6
a⁵ + b⁵ + c⁵ = 23/4 + 3abc/2
=> a⁵ + b⁵ + c⁵ = 23/4 + 1/4
=> a⁵ + b⁵ + c⁵ = 6
Hence, the value of a⁵ + b⁵ + c⁵ is 6.