Simplify fourth root of 81x^4* cube root of 27x^11

$\sqrt[4]{81 x^{4}} \cdot \sqrt[3]{27 x^{11}}$

Step 1

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

$\sqrt[4]{{(3 x)}^{4}} \cdot \sqrt[3]{27 x^{11}}$

Step 2

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

$3 x \cdot \sqrt[3]{27 x^{11}}$

Step 3

Rewrite $27 x^{11}$ as ${(3 x^{3})}^{3} x^{2}$ .

Rewrite $27$ as $3^{3}$ .

$3 x \cdot \sqrt[3]{3^{3} x^{11}}$

Factor out $x^{9}$ .

$3 x \cdot \sqrt[3]{3^{3} (x^{9} x^{2})}$

Rewrite $x^{9}$ as ${(x^{3})}^{3}$ .

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Move $x^{3}$ .

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

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

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

$3 (x^{3} x^{1}) \cdot (3 \sqrt[3]{x^{2}})$

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

$3 x^{3 + 1} \cdot (3 \sqrt[3]{x^{2}})$

Add $3$ and $1$ .

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

Step 6

Multiply $3$ by $3$ .

$9 x^{4} \sqrt[3]{x^{2}}$