Christmas Streak 68/88: Where is wrong? #1

Obviously, $√ab = √a √b$ , this is an identity, but $√a+b$ is not $√a +√b$ . It is a common mistake made in algebra.

It can't be said that $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ holds for every reals $a$ and $b.$

So the wrong part is $\boxed{\sqrt{9+16}=\sqrt{9}+\sqrt{16}}.$

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In-depth explanation:

Let's then say $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ for all reals $a$ and $b.$

Then it must be the case that $a+b=(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{a}\sqrt{b}.$

This implies that $\sqrt{a}\sqrt{b}=0$ for all reals, which is clearly false.

Therefore, for $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ to hold, at least one of $a$ and $b$ is $0.$