Simplify fourth root of 81x^5y^4* fourth root of 32x^3y

$\sqrt[4]{81 x^{5} y^{4}} \cdot \sqrt[4]{32 x^{3} y}$

Step 1

Rewrite $81 x^{5} y^{4}$ as ${(3 x y)}^{4} x$ .

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

$\sqrt[4]{3^{4} x^{5} y^{4}} \cdot \sqrt[4]{32 x^{3} y}$

Factor out $x^{4}$ .

$\sqrt[4]{3^{4} (x^{4} x) y^{4}} \cdot \sqrt[4]{32 x^{3} y}$

Move $x$ .

$\sqrt[4]{3^{4} (x^{4}) y^{4} x} \cdot \sqrt[4]{32 x^{3} y}$

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

$\sqrt[4]{{(3 x y)}^{4} x} \cdot \sqrt[4]{32 x^{3} y}$

Step 2

Pull terms out from under the radical.

$3 x y \sqrt[4]{x} \cdot \sqrt[4]{32 x^{3} y}$

Step 3

Rewrite $32 x^{3} y$ as $2^{4} \cdot (2 x^{3} y)$ .

Factor $16$ out of $32$ .

$3 x y \sqrt[4]{x} \cdot \sqrt[4]{16 (2) x^{3} y}$

Rewrite $16$ as $2^{4}$ .

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

Add parentheses.

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

Multiply $2$ by $3$ .

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

Combine using the product rule for radicals.

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

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

$6 x y \sqrt[4]{2 (x^{1} x^{3}) y}$

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

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

Add $1$ and $3$ .

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

Step 6

Rewrite $2 x^{4} y$ as $x^{4} \cdot (2 y)$ .

Reorder $2$ and $x^{4}$ .

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

Add parentheses.

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

Step 7

Pull terms out from under the radical.

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

Step 8

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Simplify fourth root of 81x^5y^4* fourth root of 32x^3y

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