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

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

Step 1

Simplify each term.

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $5$ .

$\sqrt[3]{y^{2}} (10 \sqrt[3]{y} + \sqrt[3]{y} - 9 \sqrt[3]{y^{4}})$

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

Simplify terms.

Add $10 \sqrt[3]{y}$ and $\sqrt[3]{y}$ .

$\sqrt[3]{y^{2}} (11 \sqrt[3]{y} - 9 y \sqrt[3]{y})$

Apply the distributive property.

$\sqrt[3]{y^{2}} (11 \sqrt[3]{y}) + \sqrt[3]{y^{2}} (- 9 y \sqrt[3]{y})$

Reorder.

Rewrite using the commutative property of multiplication.

$11 \sqrt[3]{y^{2}} \sqrt[3]{y} + \sqrt[3]{y^{2}} (- 9 y \sqrt[3]{y})$

Rewrite using the commutative property of multiplication.

$11 \sqrt[3]{y^{2}} \sqrt[3]{y} - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

Step 3

Simplify each term.

Multiply $11 \sqrt[3]{y^{2}} \sqrt[3]{y}$ .

Combine using the product rule for radicals.

$11 \sqrt[3]{y \cdot y^{2}} - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

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

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

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

$11 \sqrt[3]{y^{1} y^{2}} - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

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

$11 \sqrt[3]{y^{1 + 2}} - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

Add $1$ and $2$ .

$11 \sqrt[3]{y^{3}} - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

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

$11 y - 9 \sqrt[3]{y^{2}} (y \sqrt[3]{y})$

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

Combine using the product rule for radicals.

$11 y - 9 (y \sqrt[3]{y^{2} y})$

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

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

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

$11 y - 9 (y \sqrt[3]{y^{2} y^{1}})$

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

$11 y - 9 (y \sqrt[3]{y^{2 + 1}})$

Add $2$ and $1$ .

$11 y - 9 (y \sqrt[3]{y^{3}})$

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

$11 y - 9 (y \cdot y)$

Multiply $y$ by $y$ .

$11 y - 9 y^{2}$

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

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