UKMT Senior Challenge-XVII

Algebra Level 2

Given that $a+b=5$ and $ab=3$ , what is the value of $a^4+b^4$ ?

This problem is not original.This problem is part of this set .

1 solution

Thomas Siu
Mar 8, 2015

$a + b = 5$

$(a+b)^{ 2 }=5^{ 2 }$

$a^{ 2 }+2ab+b^{ 2 }=25$

Now we know $ab = 3$ , therefore $2ab = 6$ and

$a^{ 2 }+b^{ 2 }=19$

Same argument, square the above equation

$(a^{ 2 }+b^{ 2 })^{ 2 }=19^{ 2 }$

$a^{ 4 }+2(a^{ 2 } b^{ 2 })+b^{ 4 }=361$

From $ab=3$ , we can determine $(a^{ 2 }b^{ 2 })=9$

Finally $a^{ 4 }+b^{ 4 }=343$