Simplify square root of 8x^3+24x^2+18x

$\sqrt{8 x^{3} + 24 x^{2} + 18 x}$

Step 1

Factor $2 x$ out of $8 x^{3} + 24 x^{2} + 18 x$ .

Factor $2 x$ out of $8 x^{3}$ .

$\sqrt{2 x (4 x^{2}) + 24 x^{2} + 18 x}$

Factor $2 x$ out of $24 x^{2}$ .

$\sqrt{2 x (4 x^{2}) + 2 x (12 x) + 18 x}$

Factor $2 x$ out of $18 x$ .

$\sqrt{2 x (4 x^{2}) + 2 x (12 x) + 2 x (9)}$

Factor $2 x$ out of $2 x (4 x^{2}) + 2 x (12 x)$ .

$\sqrt{2 x (4 x^{2} + 12 x) + 2 x (9)}$

Factor $2 x$ out of $2 x (4 x^{2} + 12 x) + 2 x (9)$ .

$\sqrt{2 x (4 x^{2} + 12 x + 9)}$

Step 2

Factor using the perfect square rule.

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\sqrt{2 x ({(2 x)}^{2} + 12 x + 9)}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{2 x ({(2 x)}^{2} + 12 x + 3^{2})}$

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

$12 x = 2 \cdot (2 x) \cdot 3$

Rewrite the polynomial.

$\sqrt{2 x ({(2 x)}^{2} + 2 \cdot (2 x) \cdot 3 + 3^{2})}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 x$ and $b = 3$ .

$\sqrt{2 x {(2 x + 3)}^{2}}$

Step 3

Rewrite $2 x {(2 x + 3)}^{2}$ as ${(2 x + 3)}^{2} \cdot (2 x)$ .

Move $x$ .

$\sqrt{2 {(2 x + 3)}^{2} x}$

Reorder $2$ and ${(2 x + 3)}^{2}$ .

$\sqrt{{(2 x + 3)}^{2} \cdot 2 x}$

Add parentheses.

$\sqrt{{(2 x + 3)}^{2} \cdot (2 x)}$

Step 4

Pull terms out from under the radical.

$(2 x + 3) \sqrt{2 x}$