Simplify ( square root of 4x^5)( square root of 8x^4)

$(\sqrt{4 x^{5}}) (\sqrt{8 x^{4}})$

Step 1

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

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

$\sqrt{2^{2} x^{5}} \sqrt{8 x^{4}}$

Factor out $x^{4}$ .

$\sqrt{2^{2} (x^{4} x)} \sqrt{8 x^{4}}$

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

$\sqrt{2^{2} ({(x^{2})}^{2} x)} \sqrt{8 x^{4}}$

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

$\sqrt{{(2 x^{2})}^{2} x} \sqrt{8 x^{4}}$

Step 2

Pull terms out from under the radical.

$2 x^{2} \sqrt{x} \sqrt{8 x^{4}}$

Step 3

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

Factor $4$ out of $8$ .

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

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

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

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

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

Move $2$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Step 6

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

Multiply $2$ by $2$ .

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

Combine using the product rule for radicals.

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