What are the differences between: $\sqrt{(-3)^2}$, $\sqrt{-3^2}$ and $(\sqrt{-3})^2$.

Question

First, is $\sqrt{-3}$ is equal to $-3$ or is it imaginary?

What is the difference between:

• $\sqrt{(-3)^2}$
• $\sqrt{-3^2}$
• $(\sqrt{-3})^2$

Can I write $(\sqrt{-3})^2 = -3$?

And, given the rule that $\sqrt{a^n}$ is equal to ($\sqrt{a})^n$, can I say that $\sqrt{-3^2}=-3$?

Answer 1

$\sqrt{(-3)^2}=\sqrt{9}=3$,

$\sqrt{-3^2}=\sqrt{-9}=3i$,

$(\sqrt{-3})^2=(\sqrt{3}i)^2=-3$

The rule is true only in the case $a>0$.

Answer 2

$\sqrt{(-3)^2}=\sqrt{9}=3$,

$\sqrt{-3^2}=\sqrt{-9}=\pm 3i$,

$(\sqrt{-3})^2=(\sqrt{-1}\sqrt{3})^2=(\pm i\sqrt{3})^2=-3$,

$\sqrt{a^n}=(\sqrt{a})^n $ iff $ a\geq 0$,

Answer 3

The differences are in the order of operations. Parentheses are used to override the default order of operations.

Now, $\sqrt{-3}$ is most definitely NOT equal to $-3$. If $n$ is a positive integer, then $\sqrt{-n} = i \sqrt{n}$, where $i = \sqrt{-1}$ so that $i^2 = -1$. Then, since $\sqrt{3} \approx 1.732$, then $\sqrt{-3} \approx 1.732i$. On the real number line, center your compass on $0$ and put the pencil on $1.732$. Then, keeping the compass centered on $0$, move the pencil $90$ degrees.

And then:

• $\sqrt{(-3)^2} = 3$. Because of the parentheses, the first thing we do is multiply $-3$ by itself, giving us $(-3) \times (-3) = 9$, and $\sqrt{9} = 3$.
• $\sqrt{-3^2} = 3i$. Without the parentheses, we multiply $3$ by itself first (exponentiation has higher precedence than negation), giving us $9$, and then we multiply it by $-1$, so at this point we have $\sqrt{-9}$, which is $3i$.
• $(\sqrt{-3})^2 = -3$. Here the parentheses mean that we multiply $\sqrt{-3}$ by itself, and that gives us $-3$.

Lastly, if I were you, I would put that "rule" about $\sqrt{a^n}$ out of my mind, before it leads me astray any further.