Simplify cube root of y(7 cube root of 8y^2- cube root of y^5-4y cube root of 27y^2)

$\sqrt[3]{y} (7 \sqrt[3]{8 y^{2}} - \sqrt[3]{y^{5}} - 4 y \sqrt[3]{27 y^{2}})$

Simplify each term.

Rewrite $8$ as $2^{3}$ .

$\sqrt[3]{y} (7 \sqrt[3]{2^{3} y^{2}} - \sqrt[3]{y^{5}} - 4 y \sqrt[3]{27 y^{2}})$

Pull terms out from under the radical.

$\sqrt[3]{y} (7 (2 \sqrt[3]{y^{2}}) - \sqrt[3]{y^{5}} - 4 y \sqrt[3]{27 y^{2}})$

Multiply $2$ by $7$ .

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - \sqrt[3]{y^{5}} - 4 y \sqrt[3]{27 y^{2}})$

Factor out $y^{3}$ .

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - \sqrt[3]{y^{3} y^{2}} - 4 y \sqrt[3]{27 y^{2}})$

Pull terms out from under the radical.

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - (y \sqrt[3]{y^{2}}) - 4 y \sqrt[3]{27 y^{2}})$

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

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - y \sqrt[3]{y^{2}} - 4 y \sqrt[3]{3^{3} y^{2}})$

Pull terms out from under the radical.

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - y \sqrt[3]{y^{2}} - 4 y (3 \sqrt[3]{y^{2}}))$

Multiply $3$ by $- 4$ .

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - y \sqrt[3]{y^{2}} - 12 y \sqrt[3]{y^{2}})$

Simplify terms.

Subtract $12 y \sqrt[3]{y^{2}}$ from $- y \sqrt[3]{y^{2}}$ .

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}} - 13 y \sqrt[3]{y^{2}})$

Apply the distributive property.

$\sqrt[3]{y} (14 \sqrt[3]{y^{2}}) + \sqrt[3]{y} (- 13 y \sqrt[3]{y^{2}})$

Reorder.

Rewrite using the commutative property of multiplication.

$14 \sqrt[3]{y} \sqrt[3]{y^{2}} + \sqrt[3]{y} (- 13 y \sqrt[3]{y^{2}})$

Rewrite using the commutative property of multiplication.

$14 \sqrt[3]{y} \sqrt[3]{y^{2}} - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

Simplify each term.

Multiply $14 \sqrt[3]{y} \sqrt[3]{y^{2}}$ .

Combine using the product rule for radicals.

$14 \sqrt[3]{y^{2} y} - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

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

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

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

$14 \sqrt[3]{y^{2} y^{1}} - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

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

$14 \sqrt[3]{y^{2 + 1}} - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

Add $2$ and $1$ .

$14 \sqrt[3]{y^{3}} - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

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

$14 y - 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$

Multiply $- 13 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$ .

Combine using the product rule for radicals.

$14 y - 13 (y \sqrt[3]{y \cdot y^{2}})$

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

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

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

$14 y - 13 (y \sqrt[3]{y^{1} y^{2}})$

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

$14 y - 13 (y \sqrt[3]{y^{1 + 2}})$

Add $1$ and $2$ .

$14 y - 13 (y \sqrt[3]{y^{3}})$

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

$14 y - 13 (y \cdot y)$

Multiply $y$ by $y$ .

$14 y - 13 y^{2}$

Simplify cube root of y(7 cube root of 8y^2- cube root of y^5-4y cube root of 27y^2)

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