Simplify fourth root of 8x^7y^3* fourth root of 32xy^10

$\sqrt[4]{8 x^{7} y^{3}} \cdot \sqrt[4]{32 x y^{10}}$

Step 1

Rewrite $8 x^{7} y^{3}$ as $x^{4} \cdot (8 x^{3} y^{3})$ .

Factor out $x^{4}$ .

$\sqrt[4]{8 (x^{4} x^{3}) y^{3}} \cdot \sqrt[4]{32 x y^{10}}$

Reorder $8$ and $x^{4}$ .

$\sqrt[4]{x^{4} \cdot 8 x^{3} y^{3}} \cdot \sqrt[4]{32 x y^{10}}$

Add parentheses.

$\sqrt[4]{x^{4} \cdot 8 (x^{3} y^{3})} \cdot \sqrt[4]{32 x y^{10}}$

Add parentheses.

$\sqrt[4]{x^{4} \cdot (8 x^{3} y^{3})} \cdot \sqrt[4]{32 x y^{10}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{32 x y^{10}}$

Step 3

Rewrite $32 x y^{10}$ as ${(2 y^{2})}^{4} \cdot (2 x y^{2})$ .

Factor $16$ out of $32$ .

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{16 (2) x y^{10}}$

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

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{2^{4} \cdot 2 x y^{10}}$

Factor out $y^{8}$ .

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{2^{4} \cdot 2 x (y^{8} y^{2})}$

Rewrite $y^{8}$ as ${(y^{2})}^{4}$ .

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{2^{4} \cdot 2 x ({(y^{2})}^{4} y^{2})}$

Move $x$ .

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{2^{4} \cdot 2 {(y^{2})}^{4} x y^{2}}$

Move $2$ .

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{2^{4} {(y^{2})}^{4} \cdot 2 x y^{2}}$

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

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{{(2 y^{2})}^{4} \cdot 2 x y^{2}}$

Add parentheses.

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{{(2 y^{2})}^{4} \cdot 2 (x y^{2})}$

Add parentheses.

$x \sqrt[4]{8 x^{3} y^{3}} \cdot \sqrt[4]{{(2 y^{2})}^{4} \cdot (2 x y^{2})}$

Step 4

Pull terms out from under the radical.

$x \sqrt[4]{8 x^{3} y^{3}} \cdot (2 y^{2} \sqrt[4]{2 x y^{2}})$

Step 5

Rewrite using the commutative property of multiplication.

$2 x \sqrt[4]{8 x^{3} y^{3}} (y^{2} \sqrt[4]{2 x y^{2}})$

Step 6

Multiply $2 x \sqrt[4]{8 x^{3} y^{3}} (y^{2} \sqrt[4]{2 x y^{2}})$ .

Combine using the product rule for radicals.

$2 x (y^{2} \sqrt[4]{8 x^{3} y^{3} (2 x y^{2})})$

Multiply $2$ by $8$ .

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

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

$2 x (y^{2} \sqrt[4]{16 (x^{1} x^{3}) y^{3} y^{2}})$

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

$2 x (y^{2} \sqrt[4]{16 x^{1 + 3} y^{3} y^{2}})$

Add $1$ and $3$ .

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

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

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

$2 x (y^{2} \sqrt[4]{16 x^{4} y^{5}})$

Step 7

Rewrite $16 x^{4} y^{5}$ as ${(2 x y)}^{4} y$ .

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

$2 x (y^{2} \sqrt[4]{2^{4} x^{4} y^{5}})$

Factor out $y^{4}$ .

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

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 10

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

Step 11

Rewrite using the commutative property of multiplication.

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

Step 12

Multiply $2$ by $2$ .

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

Simplify fourth root of 8x^7y^3* fourth root of 32xy^10

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