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

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

Step 1

Simplify each term.

Factor out $y^{3}$ .

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

Pull terms out from under the radical.

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

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

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

Pull terms out from under the radical.

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

Multiply $3$ by $- 1$ .

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

Step 2

Simplify terms.

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

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

Apply the distributive property.

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

Step 3

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

Combine using the product rule for radicals.

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

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

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

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

$y \sqrt[3]{y^{1} y^{2}} + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

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

$y \sqrt[3]{y^{1 + 2}} + \sqrt[3]{y} (5 \sqrt[3]{y^{2}})$

Add $1$ and $2$ .

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

Step 4

Rewrite using the commutative property of multiplication.

$y \sqrt[3]{y^{3}} + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Step 5

Simplify each term.

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

$y \cdot y + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Multiply $y$ by $y$ .

$y^{2} + 5 \sqrt[3]{y} \sqrt[3]{y^{2}}$

Multiply $5 \sqrt[3]{y} \sqrt[3]{y^{2}}$ .

Combine using the product rule for radicals.

$y^{2} + 5 \sqrt[3]{y^{2} y}$

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

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

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

$y^{2} + 5 \sqrt[3]{y^{2} y^{1}}$

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

$y^{2} + 5 \sqrt[3]{y^{2 + 1}}$

Add $2$ and $1$ .

$y^{2} + 5 \sqrt[3]{y^{3}}$

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

$y^{2} + 5 y$

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

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