Simplify fourth root of 81x^8* cube root of 27x^9

$\sqrt[4]{81 x^{8}} \cdot \sqrt[3]{27 x^{9}}$

Step 1

Rewrite $81 x^{8}$ as ${(3 x^{2})}^{4}$ .

$\sqrt[4]{{(3 x^{2})}^{4}} \cdot \sqrt[3]{27 x^{9}}$

Step 2

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

$3 x^{2} \cdot \sqrt[3]{27 x^{9}}$

Step 3

Rewrite $27 x^{9}$ as ${(3 x^{3})}^{3}$ .

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

Step 4

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

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

$3 \cdot 3 x^{5}$

Step 7

Multiply $3$ by $3$ .

$9 x^{5}$