Radical Equations

$\sqrt {a - b} = 12$

$a - b = 144$

$\sqrt {a} + \sqrt {b} = 24 \ \text { (i) }$

Since (for non-negative a and b):

$a - b = ( \sqrt {a} + \sqrt {b} ) ( \sqrt {a} - \sqrt {b} )$

Therefore:

$24 ( \sqrt {a} - \sqrt {b} ) = 144$

$\sqrt {a} - \sqrt {b} = 6 \ \text { (ii) }$

(i) + (ii):

$2 \sqrt {a} = 30$

$\sqrt {a} = 15$

$15 + \sqrt {b} = 24$

$\sqrt {b} = 9$

$\sqrt {ab} = 15 × 9 = \boxed {135}$

First we have that $\sqrt{a - b} = 12 \Longrightarrow a - b = 144$ .

Next, rewrite the second equation as $\sqrt{a} = 24 - \sqrt{b}$ and then square both sides to end up with

$a = 576 - 48\sqrt{b} + b \Longrightarrow a - b = 576 - 48\sqrt{b} \Longrightarrow 144 = 576 - 48\sqrt{b}$ ,

where we substituted in our previous result $a - b = 144$ . Rearranging gives us

$576 - 144 = 48\sqrt{b} \Longrightarrow \sqrt{b} = \dfrac{432}{48} = 9 \Longrightarrow \sqrt{a} = 24 - \sqrt{b} = 15 \Longrightarrow \sqrt{ab} = \sqrt{a}\sqrt{b} = 15 \times 9 = \boxed{135}$ .