$\sqrt{ab}=\sqrt a \sqrt b$ , but $\sqrt{a+b} \neq \sqrt a + \sqrt b$

Algebra Level 2

$\large \left \{ \begin{aligned} \sqrt{a+b} & = \sqrt a+\sqrt b \\ a & \ne 0 \\ b & \ne 0 \end{aligned} \right.$

Are there any solutions to the system of equations and inequalities above?

Yes No

1 solution

Jesse Li
Feb 20, 2019

By squaring both sides of the equation, we get $a+b=a+2\sqrt{ab}+b$

Subtracting $a$ and $b$ from both sides gets us $0=2\sqrt{ab}$

This implies that $ab=0$

The only way to get 0 when multiplying 2 numbers is to have either or both numbers equal 0. However, since the inequalities tell us that $a$ and $b$ are not equal to 0, it's impossible to get 0 when multiplying them together.

Therefore, the answer is $\boxed{No}$

a b = a b \sqrt{ab}=\sqrt a \sqrt b a b ​ = a ​ b ​ , but a + b ≠ a + b \sqrt{a+b} \neq \sqrt a + \sqrt b a + b ​  ​ = a ​ + b ​

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$\sqrt{ab}=\sqrt a \sqrt b$ , but $\sqrt{a+b} \neq \sqrt a + \sqrt b$