Find the value of a*b

First, let we multiply those equation.

$\left( a+ \dfrac{1}{b} \right)\left( b+ \dfrac{1}{a} \right) = 4 \times \dfrac{16}{15}$

$ab + 2 + \dfrac{1}{ab} = \dfrac{64}{15}$

$ab + \dfrac{1}{ab} = \dfrac{34}{15}$

Let $ab = x$ , the equation becomes

$x + \dfrac{1}{x} = \dfrac{34}{15} \space \Rightarrow \text{Multiply both sides with }15x$

$15x^{2} + 15 = 34x$

$15x^{2} -34x + 15 = 0$

$(3x-5)(5x-3) = 0 \space \Rightarrow x = ab = \left( \dfrac{5}{3} , \dfrac{3}{5} \right)$

The product of all possible values of $ab$ is $\boxed{1}$

We can multiply a + 1/b = 4 and 1/a + b = 16/15 to get 2 + ab + 1/ab = 64/15 ⇒ (ab) ^2 − 34/15 (ab) + 1 = 0.

Since (34/15)^2 − 4 · 1 · 1 > 0, (b^2 - 4ac >0)

There are two roots ab, so the product of both possible values of ab is 1 by Vieta's formula for a quadratic.