Simplify the Radical Expression square root of a^3+2a^2+a

$\sqrt{a^{3} + 2 a^{2} + a}$

Step 1

Factor $a$ out of $a^{3} + 2 a^{2} + a$ .

Factor $a$ out of $a^{3}$ .

$\sqrt{a \cdot a^{2} + 2 a^{2} + a}$

Factor $a$ out of $2 a^{2}$ .

$\sqrt{a \cdot a^{2} + a (2 a) + a}$

Raise $a$ to the power of $1$ .

$\sqrt{a \cdot a^{2} + a (2 a) + a^{1}}$

Factor $a$ out of $a^{1}$ .

$\sqrt{a \cdot a^{2} + a (2 a) + a \cdot 1}$

Factor $a$ out of $a \cdot a^{2} + a (2 a)$ .

$\sqrt{a (a^{2} + 2 a) + a \cdot 1}$

Factor $a$ out of $a (a^{2} + 2 a) + a \cdot 1$ .

$\sqrt{a (a^{2} + 2 a + 1)}$

Step 2

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\sqrt{a (a^{2} + 2 a + 1^{2})}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Rewrite the polynomial.

$\sqrt{a (a^{2} + 2 \cdot a \cdot 1 + 1^{2})}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

$\sqrt{a {(a + 1)}^{2}}$

Step 3

Rewrite $a {(a + 1)}^{2}$ as ${(a + 1^{2})}^{2} a$ .

Reorder $a$ and ${(a + 1)}^{2}$ .

$\sqrt{{(a + 1)}^{2} a}$

Rewrite $1$ as $1^{2}$ .

$\sqrt{{(a + 1^{2})}^{2} a}$

Step 4

Pull terms out from under the radical.

$| a + 1^{2} | \sqrt{a}$

Step 5

One to any power is one.

$| a + 1 | \sqrt{a}$