Question

if 8a^3+b^3=14 and 2a+b=7 ,then find the value of 16a^4+b^4​

Answer 1

Step-by-step explanation:

$8 {a}^{3} + {b}^{3} = 14 \\ \\ {(2a)}^{3} + {b}^{3} = 14 \\ \\ (2a + b)( 4 {a}^{2} - 2ab + {b}^{2} ) = 14 \\ \\ 7(4 {a}^{2} - 2ab + {b}^{2} ) = 14 \\ \\ 4 {a}^{2} - 2ab + {b}^{2} = 2 \\ \\ \\ \\ \\ therefore \: \: \: \: 16 {a}^{4} + {b}^{4} = {(2a)}^{4} + {(b)}^{4} \\ \\ = {(4 {a}^{2} + {b}^{2} )}^{2} - 4 {a}^{2} {b}^{2} \\ \\ = ({(4 {a}^{2} + {b}^{2} )}^{} - 2ab)(4 {a}^{2} + {b}^{2} + 2ab) \\ \\ = 2(4 {a}^{2} + {b}^{2} + 2ab).$