Exponents and Radicals

Algebra Level 3

Under what conditions does $\sqrt{a^2+b^2}=a+b$ hold, where $a$ and $b$ are real numbers?

It is true if and only if $ab=0$ and $a+b\ge 0$. It is true if and only if $a+b\ge 0$. It is never true. It is true if and only if $ab=0$. It is always true.

2 solutions

Saya Suka
Mar 29, 2021

At least one of a or b is 0 (can also be both), and both are nonnegatives.

Fei Zhang
Mar 27, 2021

Solution 1 Square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$. Then, $0=2ab\rightarrow ab=0$. Also, it is clear that both sides of the equation must be nonnegative. The answer is $\boxed{\textbf{(D)}}$.

Solution 2 (Quick Inspection) The left side of the original equation is the arithmetic square root, which is always nonnegative. So, we need $a+b\ge 0,$ which eliminates $\textbf{(B)}$ and $\textbf{(E)}.$ Next, picking $(a,b)=(0,0)$ reveals that $\textbf{(A)}$ is incorrect, and picking $(a,b)=(1,2)$ reveals that $\textbf{(C)}$ is incorrect. By POE (Process of Elimination), the answer is $\boxed{\textbf{(D)}}.$

Solution 3 (Graphing) If we graph $\sqrt{x^2+y^2}=x+y,$ then we get the union of:

positive $x$-axis positive $y$-axis origin Therefore, the answer is $\boxed{\textbf{(D)}}.$

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Elijah L - 2 months, 2 weeks ago

Oh. Next time I'll know.

Fei Zhang - 1 month, 1 week ago

Exponents and Radicals

2 solutions

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