Simplify cube root of y(5 cube root of 8y^3- cube root of y^5-2y cube root of 64y^2)

$\sqrt[3]{y} (5 \sqrt[3]{8 y^{3}} - \sqrt[3]{y^{5}} - 2 y \sqrt[3]{64 y^{2}})$

Step 1

Simplify each term.

Rewrite $8 y^{3}$ as ${(2 y)}^{3}$ .

$\sqrt[3]{y} (5 \sqrt[3]{{(2 y)}^{3}} - \sqrt[3]{y^{5}} - 2 y \sqrt[3]{64 y^{2}})$

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

$\sqrt[3]{y} (5 (2 y) - \sqrt[3]{y^{5}} - 2 y \sqrt[3]{64 y^{2}})$

Multiply $2$ by $5$ .

$\sqrt[3]{y} (10 y - \sqrt[3]{y^{5}} - 2 y \sqrt[3]{64 y^{2}})$

Factor out $y^{3}$ .

$\sqrt[3]{y} (10 y - \sqrt[3]{y^{3} y^{2}} - 2 y \sqrt[3]{64 y^{2}})$

Pull terms out from under the radical.

$\sqrt[3]{y} (10 y - (y \sqrt[3]{y^{2}}) - 2 y \sqrt[3]{64 y^{2}})$

Rewrite $64$ as $4^{3}$ .

$\sqrt[3]{y} (10 y - y \sqrt[3]{y^{2}} - 2 y \sqrt[3]{4^{3} y^{2}})$

Pull terms out from under the radical.

$\sqrt[3]{y} (10 y - y \sqrt[3]{y^{2}} - 2 y (4 \sqrt[3]{y^{2}}))$

Multiply $4$ by $- 2$ .

$\sqrt[3]{y} (10 y - y \sqrt[3]{y^{2}} - 8 y \sqrt[3]{y^{2}})$

Step 2

Simplify terms.

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

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

Apply the distributive property.

$\sqrt[3]{y} (10 y) + \sqrt[3]{y} (- 9 y \sqrt[3]{y^{2}})$

Reorder.

Rewrite using the commutative property of multiplication.

$10 \sqrt[3]{y} y + \sqrt[3]{y} (- 9 y \sqrt[3]{y^{2}})$

Rewrite using the commutative property of multiplication.

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

Step 3

Simplify each term.

Multiply $- 9 \sqrt[3]{y} (y \sqrt[3]{y^{2}})$ .

Combine using the product rule for radicals.

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

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

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

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

$10 \sqrt[3]{y} y - 9 (y \sqrt[3]{y^{1} y^{2}})$

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

$10 \sqrt[3]{y} y - 9 (y \sqrt[3]{y^{1 + 2}})$

Add $1$ and $2$ .

$10 \sqrt[3]{y} y - 9 (y \sqrt[3]{y^{3}})$

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

$10 \sqrt[3]{y} y - 9 (y \cdot y)$

Multiply $y$ by $y$ .

$10 \sqrt[3]{y} y - 9 y^{2}$

Simplify cube root of y(5 cube root of 8y^3- cube root of y^5-2y cube root of 64y^2)

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