Simplify 3 fourth root of 27x^3*2 fourth root of 3x^5

$3 \sqrt[4]{27 x^{3}} \cdot 2 \sqrt[4]{3 x^{5}}$

Step 1

Rewrite $3 x^{5}$ as $x^{4} \cdot (3 x)$ .

Factor out $x^{4}$ .

$3 \sqrt[4]{27 x^{3}} \cdot (2 \sqrt[4]{3 (x^{4} x)})$

Reorder $3$ and $x^{4}$ .

$3 \sqrt[4]{27 x^{3}} \cdot (2 \sqrt[4]{x^{4} \cdot 3 x})$

Add parentheses.

$3 \sqrt[4]{27 x^{3}} \cdot (2 \sqrt[4]{x^{4} \cdot (3 x)})$

Step 2

Pull terms out from under the radical.

$3 \sqrt[4]{27 x^{3}} \cdot (2 (x \sqrt[4]{3 x}))$

Step 3

Multiply $3 \sqrt[4]{27 x^{3}} (2 x \sqrt[4]{3 x})$ .

Multiply $2$ by $3$ .

$6 \sqrt[4]{27 x^{3}} (x \sqrt[4]{3 x})$

Combine using the product rule for radicals.

$6 (x \sqrt[4]{27 x^{3} (3 x)})$

Multiply $3$ by $27$ .

$6 (x \sqrt[4]{81 x^{3} x})$

Raise $x$ to the power of $1$ .

$6 (x \sqrt[4]{81 (x^{1} x^{3})})$

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

$6 (x \sqrt[4]{81 x^{1 + 3}})$

Add $1$ and $3$ .

$6 (x \sqrt[4]{81 x^{4}})$

Step 4

Rewrite $81 x^{4}$ as ${(3 x)}^{4}$ .

$6 (x \sqrt[4]{{(3 x)}^{4}})$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$6 (x (3 x))$

Step 6

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$6 (x \cdot x \cdot 3)$

Multiply $x$ by $x$ .

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

Step 7

Move $3$ to the left of $x^{2}$ .

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

Step 8

Multiply $3$ by $6$ .

$18 x^{2}$