Precalculus Examples

$\sqrt[3]{x^{7}} \cdot \sqrt[3]{x^{2}}$

Step 1

Rewrite $x^{7}$ as ${(x^{2})}^{3} x$ .

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Step 1.1

Factor out $x^{6}$ .

$\sqrt[3]{x^{6} x} \cdot \sqrt[3]{x^{2}}$

Step 1.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Combine using the product rule for radicals.

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

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

Add $2$ and $1$ .

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

Step 4

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

$x^{2} x$

Step 5

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

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Step 5.1

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

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Step 5.1.1

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

$x^{2} x^{1}$

Step 5.1.2

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

$x^{2 + 1}$

Step 5.2

Add $2$ and $1$ .

$x^{3}$