Simplify square root of 32x^3- square root of 16x^3+4 square root of x^3- square root of 2x^3

$\sqrt{32 x^{3}} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Step 1

Simplify each term.

Rewrite $32 x^{3}$ as ${(4 x)}^{2} \cdot (2 x)$ .

Factor $16$ out of $32$ .

$\sqrt{16 (2) x^{3}} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2} \cdot 2 x^{3}} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Factor out $x^{2}$ .

$\sqrt{4^{2} \cdot 2 (x^{2} x)} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Move $2$ .

$\sqrt{4^{2} x^{2} \cdot 2 x} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

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

$\sqrt{{(4 x)}^{2} \cdot 2 x} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Add parentheses.

$\sqrt{{(4 x)}^{2} \cdot (2 x)} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Pull terms out from under the radical.

$4 x \sqrt{2 x} - \sqrt{16 x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Rewrite $16 x^{3}$ as ${(4 x)}^{2} x$ .

Rewrite $16$ as $4^{2}$ .

$4 x \sqrt{2 x} - \sqrt{4^{2} x^{3}} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Factor out $x^{2}$ .

$4 x \sqrt{2 x} - \sqrt{4^{2} (x^{2} x)} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

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

$4 x \sqrt{2 x} - \sqrt{{(4 x)}^{2} x} + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Pull terms out from under the radical.

$4 x \sqrt{2 x} - (4 x \sqrt{x}) + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Multiply $4$ by $- 1$ .

$4 x \sqrt{2 x} - 4 (x \sqrt{x}) + 4 \sqrt{x^{3}} - \sqrt{2 x^{3}}$

Factor out $x^{2}$ .

$4 x \sqrt{2 x} - 4 x \sqrt{x} + 4 \sqrt{x^{2} x} - \sqrt{2 x^{3}}$

Pull terms out from under the radical.

$4 x \sqrt{2 x} - 4 x \sqrt{x} + 4 (x \sqrt{x}) - \sqrt{2 x^{3}}$

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

Factor out $x^{2}$ .

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

Reorder $2$ and $x^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

$4 x \sqrt{2 x} - 4 x \sqrt{x} + 4 x \sqrt{x} - x \sqrt{2 x}$

Step 2

Simplify by adding terms.

Combine the opposite terms in $4 x \sqrt{2 x} - 4 x \sqrt{x} + 4 x \sqrt{x} - x \sqrt{2 x}$ .

Add $- 4 x \sqrt{x}$ and $4 x \sqrt{x}$ .

$4 x \sqrt{2 x} + 0 - x \sqrt{2 x}$

Add $4 x \sqrt{2 x}$ and $0$ .

$4 x \sqrt{2 x} - x \sqrt{2 x}$

Subtract $x \sqrt{2 x}$ from $4 x \sqrt{2 x}$ .

$3 x \sqrt{2 x}$

Simplify square root of 32x^3- square root of 16x^3+4 square root of x^3- square root of 2x^3

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