Simplify the Radical Expression -4x square root of 125x^2+5 square root of 125x^4

$- 4 x \sqrt{125 x^{2}} + 5 \sqrt{125 x^{4}}$

Step 1

Simplify each term.

Rewrite $125 x^{2}$ as ${(5 x)}^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$- 4 x \sqrt{25 (5) x^{2}} + 5 \sqrt{125 x^{4}}$

Rewrite $25$ as $5^{2}$ .

$- 4 x \sqrt{5^{2} \cdot 5 x^{2}} + 5 \sqrt{125 x^{4}}$

Move $5$ .

$- 4 x \sqrt{5^{2} x^{2} \cdot 5} + 5 \sqrt{125 x^{4}}$

Rewrite $5^{2} x^{2}$ as ${(5 x)}^{2}$ .

$- 4 x \sqrt{{(5 x)}^{2} \cdot 5} + 5 \sqrt{125 x^{4}}$

Pull terms out from under the radical.

$- 4 x (| 5 x | \sqrt{5}) + 5 \sqrt{125 x^{4}}$

Remove non-negative terms from the absolute value.

$- 4 x (5 | x | \sqrt{5}) + 5 \sqrt{125 x^{4}}$

Multiply $5$ by $- 4$ .

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{125 x^{4}}$

Rewrite $125 x^{4}$ as ${(5 x^{2})}^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{25 (5) x^{4}}$

Rewrite $25$ as $5^{2}$ .

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{5^{2} \cdot 5 x^{4}}$

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

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{5^{2} \cdot 5 {(x^{2})}^{2}}$

Move $5$ .

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{5^{2} {(x^{2})}^{2} \cdot 5}$

Rewrite $5^{2} {(x^{2})}^{2}$ as ${(5 x^{2})}^{2}$ .

$- 20 x (| x | \sqrt{5}) + 5 \sqrt{{(5 x^{2})}^{2} \cdot 5}$

Pull terms out from under the radical.

$- 20 x (| x | \sqrt{5}) + 5 (| 5 x^{2} | \sqrt{5})$

Remove non-negative terms from the absolute value.

$- 20 x (| x | \sqrt{5}) + 5 (5 x^{2} \sqrt{5})$

Multiply $5$ by $5$ .

$- 20 x (| x | \sqrt{5}) + 25 x^{2} \sqrt{5}$

Step 2

Reorder factors in $- 20 x (| x | \sqrt{5}) + 25 x^{2} \sqrt{5}$ .

$- 20 x | x | \sqrt{5} + 25 x^{2} \sqrt{5}$