Simplify 4 square root of 6x*5 square root of 8x^4

$4 \sqrt{6 x} \cdot 5 \sqrt{8 x^{4}}$

Step 1

Rewrite $8 x^{4}$ as ${(2 x^{2})}^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$4 \sqrt{6 x} \cdot (5 \sqrt{4 (2) x^{4}})$

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

$4 \sqrt{6 x} \cdot (5 \sqrt{2^{2} \cdot 2 x^{4}})$

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

$4 \sqrt{6 x} \cdot (5 \sqrt{2^{2} \cdot 2 {(x^{2})}^{2}})$

Move $2$ .

$4 \sqrt{6 x} \cdot (5 \sqrt{2^{2} {(x^{2})}^{2} \cdot 2})$

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

$4 \sqrt{6 x} \cdot (5 \sqrt{{(2 x^{2})}^{2} \cdot 2})$

Step 2

Pull terms out from under the radical.

$4 \sqrt{6 x} \cdot (5 (2 x^{2} \sqrt{2}))$

Step 3

Multiply $2$ by $5$ .

$4 \sqrt{6 x} \cdot (10 (x^{2} \sqrt{2}))$

Step 4

Multiply $4 \sqrt{6 x} (10 x^{2} \sqrt{2})$ .

Multiply $10$ by $4$ .

$40 \sqrt{6 x} (x^{2} \sqrt{2})$

Combine using the product rule for radicals.

$40 (x^{2} \sqrt{6 x \cdot 2})$

Multiply $2$ by $6$ .

$40 (x^{2} \sqrt{12 x})$

Step 5

Rewrite $12 x$ as $2^{2} \cdot (3 x)$ .

Factor $4$ out of $12$ .

$40 (x^{2} \sqrt{4 (3) x})$

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

$40 (x^{2} \sqrt{2^{2} \cdot 3 x})$

Add parentheses.

$40 (x^{2} \sqrt{2^{2} \cdot (3 x)})$

Step 6

Pull terms out from under the radical.

$40 (x^{2} (2 \sqrt{3 x}))$

Step 7

Simplify the expression.

Rewrite using the commutative property of multiplication.

$40 (2 x^{2} \sqrt{3 x})$

Multiply $2$ by $40$ .

$80 x^{2} \sqrt{3 x}$