Simplify cube root of y^2(5 cube root of y+ cube root of y^4-9 cube root of 8y)

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + \sqrt[3]{y^{4}} - 9 \sqrt[3]{8 y})$

Simplify each term.

Factor out $y^{3}$ .

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + \sqrt[3]{y^{3} y} - 9 \sqrt[3]{8 y})$

Pull terms out from under the radical.

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + y \sqrt[3]{y} - 9 \sqrt[3]{8 y})$

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

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + y \sqrt[3]{y} - 9 \sqrt[3]{2^{3} y})$

Pull terms out from under the radical.

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + y \sqrt[3]{y} - 9 (2 \sqrt[3]{y}))$

Multiply $2$ by $- 9$ .

$\sqrt[3]{y^{2}} (5 \sqrt[3]{y} + y \sqrt[3]{y} - 18 \sqrt[3]{y})$

Simplify terms.

Subtract $18 \sqrt[3]{y}$ from $5 \sqrt[3]{y}$ .

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

Apply the distributive property.

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

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

Multiply $y$ by $y$ .

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

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

Combine using the product rule for radicals.

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

$y^{2} - 13 y$

Simplify cube root of y^2(5 cube root of y+ cube root of y^4-9 cube root of 8y)

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