Simplify square root of 9x^4* cube root of 27x^10

$\sqrt{9 x^{4}} \cdot \sqrt[3]{27 x^{10}}$

Step 1

Rewrite $9 x^{4}$ as ${(3 x^{2})}^{2}$ .

$\sqrt{{(3 x^{2})}^{2}} \cdot \sqrt[3]{27 x^{10}}$

Step 2

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

$3 x^{2} \cdot \sqrt[3]{27 x^{10}}$

Step 3

Rewrite $27 x^{10}$ as ${(3 x^{3})}^{3} x$ .

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

$3 x^{2} \cdot \sqrt[3]{3^{3} x^{10}}$

Factor out $x^{9}$ .

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

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

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

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

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Move $x^{3}$ .

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

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

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

Add $3$ and $2$ .

$3 x^{5} \cdot (3 \sqrt[3]{x})$

Step 6

Multiply $3$ by $3$ .

$9 x^{5} \sqrt[3]{x}$