6th Power

Algebra Level 3

$\large \frac{a}{a+b^2} + \frac{b}{b+a^2} = \frac{7}{8}$

Let $a$ and $b$ be positive real numbers such that their product is $2$ and satisfy the equation above.

What is the value of $a^6+b^6$ ?

1 solution

Fahim Shahriar Shakkhor
Mar 2, 2015

$\frac{a}{a+b^2} + \frac{b}{b+a^2} = \frac{7}{8}$

$\Rightarrow \frac{a(b+a^2)+b(a+b^2)}{(b+a^2)(a+b^2)} = \frac{7}{8}$

$\Rightarrow \frac{2ab+a^3+b^3}{a^3+b^3+ab+a^2b^2} = \frac{7}{8}$

$\Rightarrow \frac{4+a^3+b^3}{a^3+b^3+6} = \frac{7}{8}\quad [ab=2]$

$\Rightarrow a^3 + b^3 = 10$

$\Rightarrow (a^3 + b^3)^2 = 10^2$

$\Rightarrow a^6 + b^6 + 2a^3b^3 = 10^2$

$\Rightarrow a^6 + b^6 = 100 - 16 = \boxed{84}$

Used exactly the same solution :)

Curtis Clement - 6 years, 3 months ago