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

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

Step 1

Simplify each term.

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 4$ .

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

Step 2

Simplify terms.

Subtract $y \sqrt[3]{y^{2}}$ from $4 y \sqrt[3]{y^{2}}$ .

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

Apply the distributive property.

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

Reorder.

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

Step 3

Simplify each term.

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

Combine using the product rule for radicals.

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

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

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

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

$3 (y \sqrt[3]{y^{1} y^{2}}) - 8 \sqrt[3]{y} \sqrt[3]{y^{2}}$

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

$3 (y \sqrt[3]{y^{1 + 2}}) - 8 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Add $1$ and $2$ .

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

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

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

Multiply $y$ by $y$ .

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

Multiply $- 8 \sqrt[3]{y} \sqrt[3]{y^{2}}$ .

Combine using the product rule for radicals.

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

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

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

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

$3 y^{2} - 8 \sqrt[3]{y^{2} y^{1}}$

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

$3 y^{2} - 8 \sqrt[3]{y^{2 + 1}}$

Add $2$ and $1$ .

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

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

$3 y^{2} - 8 y$

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

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