Simplify cube root of x( cube root of 81x^2- cube root of x)

$\sqrt[3]{x} (\sqrt[3]{81 x^{2}} - \sqrt[3]{x})$

Step 1

Simplify each term.

Rewrite $81 x^{2}$ as $3^{3} \cdot (3 x^{2})$ .

Factor $27$ out of $81$ .

$\sqrt[3]{x} (\sqrt[3]{27 (3) x^{2}} - \sqrt[3]{x})$

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Step 2

Simplify by multiplying through.

Apply the distributive property.

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

Reorder.

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

$3 \sqrt[3]{x} \sqrt[3]{3 x^{2}} - \sqrt[3]{x} \sqrt[3]{x}$

Step 3

Simplify each term.

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

Combine using the product rule for radicals.

$3 \sqrt[3]{3 x^{2} x} - \sqrt[3]{x} \sqrt[3]{x}$

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

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

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

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

Add $1$ and $2$ .

$3 \sqrt[3]{3 x^{3}} - \sqrt[3]{x} \sqrt[3]{x}$

Reorder $3$ and $x^{3}$ .

$3 \sqrt[3]{x^{3} \cdot 3} - \sqrt[3]{x} \sqrt[3]{x}$

Pull terms out from under the radical.

$3 (x \sqrt[3]{3}) - \sqrt[3]{x} \sqrt[3]{x}$

Multiply $- \sqrt[3]{x} \sqrt[3]{x}$ .

Raise $\sqrt[3]{x}$ to the power of $1$ .

$3 x \sqrt[3]{3} - ({\sqrt[3]{x}}^{1} \sqrt[3]{x})$

Raise $\sqrt[3]{x}$ to the power of $1$ .

$3 x \sqrt[3]{3} - ({\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{1})$

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

$3 x \sqrt[3]{3} - {\sqrt[3]{x}}^{1 + 1}$

Add $1$ and $1$ .

$3 x \sqrt[3]{3} - {\sqrt[3]{x}}^{2}$

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$3 x \sqrt[3]{3} - \sqrt[3]{x^{2}}$

Simplify cube root of x( cube root of 81x^2- cube root of x)

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