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

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

Step 1

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

Combine using the product rule for radicals.

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

Multiply $27$ by $3$ .

$\sqrt[4]{81 x^{2} x^{3}}$

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

Move $x^{3}$ .

$\sqrt[4]{81 (x^{3} x^{2})}$

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

$\sqrt[4]{81 x^{3 + 2}}$

Add $3$ and $2$ .

$\sqrt[4]{81 x^{5}}$

Step 2

Rewrite $81 x^{5}$ as ${(3 x)}^{4} x$ .

Rewrite $81$ as $3^{4}$ .

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

Factor out $x^{4}$ .

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

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

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

Step 3

Pull terms out from under the radical.

$3 x \sqrt[4]{x}$