Simplify (6 cube root of xy^2z^0)( fourth root of 81x^3y)

$(6 \sqrt[3]{x y^{2} z^{0}}) (\sqrt[4]{81 x^{3} y})$

Anything raised to $0$ is $1$ .

$6 \sqrt[3]{x y^{2} \cdot 1} \sqrt[4]{81 x^{3} y}$

Multiply $x$ by $1$ .

$6 \sqrt[3]{x y^{2}} \sqrt[4]{81 x^{3} y}$

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

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

$6 \sqrt[3]{x y^{2}} \sqrt[4]{3^{4} x^{3} y}$

Add parentheses.

$6 \sqrt[3]{x y^{2}} \sqrt[4]{3^{4} (x^{3} y)}$

Pull terms out from under the radical.

$6 \sqrt[3]{x y^{2}} (3 \sqrt[4]{x^{3} y})$

Multiply $6 \sqrt[3]{x y^{2}} (3 \sqrt[4]{x^{3} y})$ .

Multiply $3$ by $6$ .

$18 \sqrt[3]{x y^{2}} \sqrt[4]{x^{3} y}$

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3} y}$ as ${(x^{3} y)}^{\frac{1}{4}}$ .

$18 ({(x^{3} y)}^{\frac{1}{4}} \sqrt[3]{x y^{2}})$

Rewrite ${(x^{3} y)}^{\frac{1}{4}}$ as ${(x^{3} y)}^{\frac{3}{12}}$ .

$18 ({(x^{3} y)}^{\frac{3}{12}} \sqrt[3]{x y^{2}})$

Rewrite ${(x^{3} y)}^{\frac{3}{12}}$ as $\sqrt[12]{{(x^{3} y)}^{3}}$ .

$18 (\sqrt[12]{{(x^{3} y)}^{3}} \sqrt[3]{x y^{2}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x y^{2}}$ as ${(x y^{2})}^{\frac{1}{3}}$ .

$18 (\sqrt[12]{{(x^{3} y)}^{3}} {(x y^{2})}^{\frac{1}{3}})$

Rewrite ${(x y^{2})}^{\frac{1}{3}}$ as ${(x y^{2})}^{\frac{4}{12}}$ .

$18 (\sqrt[12]{{(x^{3} y)}^{3}} {(x y^{2})}^{\frac{4}{12}})$

Rewrite ${(x y^{2})}^{\frac{4}{12}}$ as $\sqrt[12]{{(x y^{2})}^{4}}$ .

$18 (\sqrt[12]{{(x^{3} y)}^{3}} \sqrt[12]{{(x y^{2})}^{4}})$

Combine using the product rule for radicals.

$18 \sqrt[12]{{(x^{3} y)}^{3} {(x y^{2})}^{4}}$

Apply the product rule to $x^{3} y$ .

$18 \sqrt[12]{{(x^{3})}^{3} y^{3} {(x y^{2})}^{4}}$

Apply the product rule to $x y^{2}$ .

$18 \sqrt[12]{{(x^{3})}^{3} y^{3} x^{4} {(y^{2})}^{4}}$

Multiply the exponents in ${(x^{3})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$18 \sqrt[12]{x^{3 \cdot 3} y^{3} x^{4} {(y^{2})}^{4}}$

Multiply $3$ by $3$ .

$18 \sqrt[12]{x^{9} y^{3} x^{4} {(y^{2})}^{4}}$

Multiply the exponents in ${(y^{2})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$18 \sqrt[12]{x^{9} y^{3} x^{4} y^{2 \cdot 4}}$

Multiply $2$ by $4$ .

$18 \sqrt[12]{x^{9} y^{3} x^{4} y^{8}}$

Combine exponents.

Multiply $x^{9}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$18 \sqrt[12]{x^{4} x^{9} y^{3} y^{8}}$

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

$18 \sqrt[12]{x^{4 + 9} y^{3} y^{8}}$

Add $4$ and $9$ .

$18 \sqrt[12]{x^{13} y^{3} y^{8}}$

Multiply $y^{3}$ by $y^{8}$ by adding the exponents.

Move $y^{8}$ .

$18 \sqrt[12]{x^{13} (y^{8} y^{3})}$

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

$18 \sqrt[12]{x^{13} y^{8 + 3}}$

Add $8$ and $3$ .

$18 \sqrt[12]{x^{13} y^{11}}$

Rewrite $x^{13} y^{11}$ as $x^{12} (x y^{11})$ .

Factor out $x^{12}$ .

$18 \sqrt[12]{x^{12} x y^{11}}$

Add parentheses.

$18 \sqrt[12]{x^{12} (x y^{11})}$

Pull terms out from under the radical.

$18 x \sqrt[12]{x y^{11}}$

Simplify (6 cube root of xy^2z^0)( fourth root of 81x^3y)

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