Simplify 4x^3 square root of 8x^7y^3

$4 x^{3} \sqrt{8 x^{7} y^{3}}$

Step 1

Rewrite $8 x^{7} y^{3}$ as ${(2 x^{3} y)}^{2} \cdot (2 x y)$ .

Factor $4$ out of $8$ .

$4 x^{3} \sqrt{4 (2) x^{7} y^{3}}$

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

$4 x^{3} \sqrt{2^{2} \cdot 2 x^{7} y^{3}}$

Factor out $x^{6}$ .

$4 x^{3} \sqrt{2^{2} \cdot 2 (x^{6} x) y^{3}}$

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$4 x^{3} \sqrt{2^{2} \cdot 2 ({(x^{3})}^{2} x) y^{3}}$

Factor out $y^{2}$ .

$4 x^{3} \sqrt{2^{2} \cdot 2 ({(x^{3})}^{2} x) (y^{2} y)}$

Move $x$ .

$4 x^{3} \sqrt{2^{2} \cdot 2 {(x^{3})}^{2} y^{2} x y}$

Move $2$ .

$4 x^{3} \sqrt{2^{2} {(x^{3})}^{2} y^{2} \cdot 2 x y}$

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

$4 x^{3} \sqrt{{(2 x^{3} y)}^{2} \cdot 2 x y}$

Add parentheses.

$4 x^{3} \sqrt{{(2 x^{3} y)}^{2} \cdot 2 (x y)}$

Add parentheses.

$4 x^{3} \sqrt{{(2 x^{3} y)}^{2} \cdot (2 x y)}$

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $3$ and $3$ .

$4 x^{6} (2 y \sqrt{2 x y})$

Step 4

Multiply $2$ by $4$ .

$8 x^{6} (y \sqrt{2 x y})$

Step 5

Reorder factors in $8 x^{6} (y \sqrt{2 x y})$ .

$8 x^{6} y \sqrt{2 x y}$