# How do you simplify sqrt((a +2)^2)?

Aug 30, 2017

$a + 2$

#### Explanation:

$\sqrt{{\left(a + 2\right)}^{2}}$ means to take the square root of ${\left(a + 2\right)}^{2}$, which is $a + 2$.

Aug 30, 2017

$\sqrt{{\left(a + 2\right)}^{2}} = a + 2$$\text{ } a \in \left[- 2 , - \infty\right)$
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$\sqrt{{\left(a + 2\right)}^{2}} = - \left(a + 2\right)$ $\text{ } a \in \left(- \infty , - 2\right)$

#### Explanation:

$\sqrt{{\left(a + 2\right)}^{2}} = \left\mid a + 2 \right\mid$
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If$\text{ "color (blue)(a+2>=0)rArra >= -2" }$
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then
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$\sqrt{{\left(a + 2\right)}^{2}} = \textcolor{b l u e}{a + 2}$ $\text{ } a \in \left[- 2 , - \infty\right)$
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If$\text{ "color (red)(a+2 < 0 ) rArr a <-2" }$
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then
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$\sqrt{{\left(a + 2\right)}^{2}} = \textcolor{red}{- \left(a + 2\right)}$ $\text{ } a \in \left(- \infty , - 2\right)$

Aug 31, 2017

$\sqrt{{\left(a + 2\right)}^{2}} = \left\mid a + 2 \right\mid$

#### Explanation:

A square root of a number $x$ is a number $y$ such that ${y}^{2} = x$.

Any non-zero number $x$ has two square roots, which we write as $\sqrt{x}$ and $- \sqrt{x}$. The principal square root is $\sqrt{x}$.

If $x$ is positive then $\sqrt{x}$ is the positive square root and $- \sqrt{x}$ the negative one.

If $x = {t}^{2}$ for some number $t$ then the square roots of $x$ are $t$ and $- t$.

Hence we find that the square roots of ${\left(a + 2\right)}^{2}$ are $\left(a + 2\right)$ and $- \left(a + 2\right)$.

Which of $\left(a + 2\right)$ and $- \left(a + 2\right)$ is the principal, non-negative one? Whichever is positive, or if zero, then they are both the same.

We can automatically choose between the two using the absolute value and write:

$\sqrt{{\left(a + 2\right)}^{2}} = \left\mid a + 2 \right\mid$