Simplify fourth root of 81x^12* square root of 81x

$\sqrt[4]{81 x^{12}} \cdot \sqrt{81 x}$

Rewrite $81 x^{12}$ as ${(3 x^{3})}^{4}$ .

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

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

$3 x^{3} \cdot \sqrt{81 x}$

Rewrite $81 x$ as ${(3^{2})}^{2} x$ .

Rewrite $81$ as $9^{2}$ .

$3 x^{3} \cdot \sqrt{9^{2} x}$

Rewrite $9$ as $3^{2}$ .

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

Pull terms out from under the radical.

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

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

Move $3^{2}$ .

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

Multiply $3^{2}$ by $3$ .

Raise $3$ to the power of $1$ .

$3^{2} \cdot 3^{1} x^{3} \cdot \sqrt{x}$

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

$3^{2 + 1} x^{3} \cdot \sqrt{x}$

Add $2$ and $1$ .

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

Raise $3$ to the power of $3$ .

$27 x^{3} \sqrt{x}$