Algebra Examples

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + \sqrt[3]{y^{5}} - \sqrt[3]{27 y^{2}})$

Step 1

Simplify each term.

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Step 1.1

Factor out $y^{3}$ .

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + \sqrt[3]{y^{3} y^{2}} - \sqrt[3]{27 y^{2}})$

Step 1.2

Pull terms out from under the radical.

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + y \sqrt[3]{y^{2}} - \sqrt[3]{27 y^{2}})$

Step 1.3

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

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + y \sqrt[3]{y^{2}} - \sqrt[3]{3^{3} y^{2}})$

Step 1.4

Pull terms out from under the radical.

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + y \sqrt[3]{y^{2}} - (3 \sqrt[3]{y^{2}}))$

Step 1.5

Multiply $3$ by $- 1$ .

$\sqrt[3]{y} (8 \sqrt[3]{y^{2}} + y \sqrt[3]{y^{2}} - 3 \sqrt[3]{y^{2}})$

Step 2

Simplify terms.

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Step 2.1

Subtract $3 \sqrt[3]{y^{2}}$ from $8 \sqrt[3]{y^{2}}$ .

$\sqrt[3]{y} (y \sqrt[3]{y^{2}} + 5 \sqrt[3]{y^{2}})$

Step 2.2

Apply the distributive property.

$\sqrt[3]{y} (y \sqrt[3]{y^{2}}) + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

Step 3

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

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Step 3.1

Combine using the product rule for radicals.

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

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

$y \sqrt[3]{y^{1} y^{2}} + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

Step 3.2.1.2

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

$y \sqrt[3]{y^{1 + 2}} + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

Step 3.2.2

Add $1$ and $2$ .

$y \sqrt[3]{y^{3}} + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

Step 4

Rewrite using the commutative property of multiplication.

$y \sqrt[3]{y^{3}} + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Step 5

Simplify each term.

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Step 5.1

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

$y \cdot y + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Step 5.2

Multiply $y$ by $y$ .

$y^{2} + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Step 5.3

Multiply $5 \sqrt[3]{y} \sqrt[3]{y^{2}}$ .

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Step 5.3.1

Combine using the product rule for radicals.

$y^{2} + 5 \sqrt[3]{y^{2} y}$

Step 5.3.2

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

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Step 5.3.2.1

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

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Step 5.3.2.1.1

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

$y^{2} + 5 \sqrt[3]{y^{2} y^{1}}$

Step 5.3.2.1.2

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

$y^{2} + 5 \sqrt[3]{y^{2 + 1}}$

Step 5.3.2.2

Add $2$ and $1$ .

$y^{2} + 5 \sqrt[3]{y^{3}}$

Step 5.4

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

$y^{2} + 5 y$