Calculus Examples

$\sqrt[3]{81 x^{5} y^{10}} \sqrt[3]{9 x y^{2}}$

Step 1

Rewrite $81 x^{5} y^{10}$ as ${(3 x y^{3})}^{3} \cdot (3 x^{2} y)$ .

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Step 1.1

Factor $27$ out of $81$ .

$\sqrt[3]{27 (3) x^{5} y^{10}} \sqrt[3]{9 x y^{2}}$

Step 1.2

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

$\sqrt[3]{3^{3} \cdot 3 x^{5} y^{10}} \sqrt[3]{9 x y^{2}}$

Step 1.3

Factor out $x^{3}$ .

$\sqrt[3]{3^{3} \cdot 3 (x^{3} x^{2}) y^{10}} \sqrt[3]{9 x y^{2}}$

Step 1.4

Factor out $y^{9}$ .

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

Step 1.5

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

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

Step 1.6

Move $x^{2}$ .

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

Step 1.7

Move $3$ .

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

Step 1.8

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

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

Step 1.9

Add parentheses.

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

Step 1.10

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$3 x y^{3} \sqrt[3]{3 x^{2} y} \sqrt[3]{9 x y^{2}}$

Step 3

Multiply $3 x y^{3} \sqrt[3]{3 x^{2} y} \sqrt[3]{9 x y^{2}}$ .

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Step 3.1

Combine using the product rule for radicals.

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

Step 3.2

Multiply $3$ by $9$ .

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

Step 3.3

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

$3 x y^{3} \sqrt[3]{27 (x^{2} x^{1}) y^{2} y}$

Step 3.4

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

$3 x y^{3} \sqrt[3]{27 x^{2 + 1} y^{2} y}$

Step 3.5

Add $2$ and $1$ .

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

Step 3.6

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

$3 x y^{3} \sqrt[3]{27 x^{3} (y^{1} y^{2})}$

Step 3.7

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

$3 x y^{3} \sqrt[3]{27 x^{3} y^{1 + 2}}$

Step 3.8

Add $1$ and $2$ .

$3 x y^{3} \sqrt[3]{27 x^{3} y^{3}}$

Step 4

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

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

Step 5

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

$3 x y^{3} (3 x y)$

Step 6

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

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Step 6.1

Move $x$ .

$3 (x \cdot x) y^{3} (3 y)$

Step 6.2

Multiply $x$ by $x$ .

$3 x^{2} y^{3} (3 y)$

Step 7

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

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Step 7.1

Move $y$ .

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

Step 7.2

Multiply $y$ by $y^{3}$ .

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Step 7.2.1

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

$3 x^{2} (y^{1} y^{3}) \cdot 3$

Step 7.2.2

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

$3 x^{2} y^{1 + 3} \cdot 3$

Step 7.3

Add $1$ and $3$ .

$3 x^{2} y^{4} \cdot 3$

Step 8

Multiply $3$ by $3$ .

$9 x^{2} y^{4}$