How can we simplify this expression involving square roots?

[Zoey323]

\(\displaystyle 4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}\)
I don't even know where to start, I know the teacher said to distribute but distibute what?

 

[topsquark]

\(\displaystyle 4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}\)
I don't even know where to start, I know the teacher said to distribute but distibute what?

Well, you could start by simplifying \(\displaystyle \sqrt{n^2}\). What is that?

-Dan

 

[Amad27]

\(\displaystyle 4\sqrt{n^{2}}+\sqrt{m^{2}n-\sqrt{4n^{2}}}-\sqrt{mn^{2}}\)
I don't even know where to start, I know the teacher said to distribute but distibute what?

Not necessarily distribute but observe and simplify.

Use facts like

$\sqrt(n^2) = n$ and facts like $\sqrt(4) = 2$

To start you off, notice,

$4\sqrt(n^2) = 4n$

Also remember facts like

$\sqrt(ab) = \sqrt(a)\cdot\sqrt(b)$

 

[MarkFL]

If $n$ is known to be non-negative, then we can write:

\(\displaystyle \sqrt{n^2}=n\)

Otherwise, we should write by definition:

\(\displaystyle \sqrt{n^2}=|n|\)

 

[CellCoree]

To simplify this expression, we can first use the properties of square roots to rewrite it as 4n + √(m²n - 2n) - √(mn²). Then, we can distribute the √ symbol to each term inside the parentheses, resulting in 4n + √(m²n) - √(2n) - √(mn²). Using the property of square roots, we can simplify √(m²n) to mn and √(mn²) to √(n²m²) = nm. This leaves us with the simplified expression of 4n + mn - √(2n) - nm.