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

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

Step 1

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

Factor out $x^{6}$ .

$\sqrt{3 (x^{6} x)} \cdot \sqrt{8 x^{4}}$

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

$\sqrt{3 ({(x^{3})}^{2} x)} \cdot \sqrt{8 x^{4}}$

Reorder $3$ and ${(x^{3})}^{2}$ .

$\sqrt{{(x^{3})}^{2} \cdot 3 x} \cdot \sqrt{8 x^{4}}$

Add parentheses.

$\sqrt{{(x^{3})}^{2} \cdot (3 x)} \cdot \sqrt{8 x^{4}}$

Step 2

Pull terms out from under the radical.

$x^{3} \sqrt{3 x} \cdot \sqrt{8 x^{4}}$

Step 3

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

Factor $4$ out of $8$ .

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

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

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

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

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

Move $2$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

Add $2$ and $3$ .

$2 \sqrt{2} x^{5} \sqrt{3 x}$

Step 7

Multiply $2 \sqrt{2} x^{5} \sqrt{3 x}$ .

Combine using the product rule for radicals.

$2 x^{5} \sqrt{2 (3 x)}$

Multiply $2$ by $3$ .

$2 x^{5} \sqrt{6 x}$

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

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