Finite Math Examples

$\sqrt{2 x^{3} y} \sqrt[4]{8 x y^{7}}$

Step 1

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

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Step 1.1

Factor out $x^{2}$ .

$\sqrt{2 (x^{2} x) y} \sqrt[4]{8 x y^{7}}$

Step 1.2

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

$\sqrt{x^{2} \cdot 2 x y} \sqrt[4]{8 x y^{7}}$

Step 1.3

Add parentheses.

$\sqrt{x^{2} \cdot 2 (x y)} \sqrt[4]{8 x y^{7}}$

Step 1.4

Add parentheses.

$\sqrt{x^{2} \cdot (2 x y)} \sqrt[4]{8 x y^{7}}$

Step 2

Pull terms out from under the radical.

$x \sqrt{2 x y} \sqrt[4]{8 x y^{7}}$

Step 3

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

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Step 3.1

Factor out $y^{4}$ .

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

Step 3.2

Move $x$ .

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

Step 3.3

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

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

Step 3.4

Add parentheses.

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

Step 3.5

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Rewrite the expression using the least common index of $4$ .

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Step 5.1.1

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

$x (y ({(2 x y)}^{\frac{1}{2}} \sqrt[4]{8 x y^{3}}))$

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Combine using the product rule for radicals.

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

Step 5.3

Apply the product rule to $2 x y$ .

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

Step 5.4

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

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Step 5.4.1

Move $y^{3}$ .

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

Step 5.4.2

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

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

Step 5.4.3

Add $3$ and $2$ .

$x (y \sqrt[4]{{(2 x)}^{2} y^{5} \cdot 8 x})$

Step 5.5

Apply the product rule to $2 x$ .

$x (y \sqrt[4]{2^{2} x^{2} y^{5} \cdot 8 x})$

Step 5.6

Raise $2$ to the power of $2$ .

$x (y \sqrt[4]{4 x^{2} y^{5} \cdot 8 x})$

Step 5.7

Combine exponents.

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Step 5.7.1

Multiply $8$ by $4$ .

$x (y \sqrt[4]{32 x^{2} y^{5} x})$

Step 5.7.2

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

$x (y \sqrt[4]{32 (x^{1} x^{2}) y^{5}})$

Step 5.7.3

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

$x (y \sqrt[4]{32 x^{1 + 2} y^{5}})$

Step 5.7.4

Add $1$ and $2$ .

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

Step 6

Rewrite $32 x^{3} y^{5}$ as ${(2 y)}^{4} \cdot (2 x^{3} y)$ .

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Step 6.1

Factor $16$ out of $32$ .

$x (y \sqrt[4]{16 (2) x^{3} y^{5}})$

Step 6.2

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

$x (y \sqrt[4]{2^{4} \cdot 2 x^{3} y^{5}})$

Step 6.3

Factor out $y^{4}$ .

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

Step 6.4

Move $x^{3}$ .

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

Step 6.5

Move $2$ .

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

Step 6.6

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

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

Step 6.7

Add parentheses.

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

Step 6.8

Add parentheses.

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

Step 7

Pull terms out from under the radical.

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

Step 8

Simplify the expression.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Multiply $y$ by $y$ .

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