Precalculus Examples

$\sqrt[3]{4 x^{3} z^{2}} \sqrt[3]{4 x^{5} z^{4}}$

Step 1

Rewrite $4 x^{3} z^{2}$ as $x^{3} \cdot (4 z^{2})$ .

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Step 1.1

Reorder $4$ and $x^{3}$ .

$\sqrt[3]{x^{3} \cdot 4 z^{2}} \sqrt[3]{4 x^{5} z^{4}}$

Step 1.2

Add parentheses.

$\sqrt[3]{x^{3} \cdot (4 z^{2})} \sqrt[3]{4 x^{5} z^{4}}$

Step 2

Pull terms out from under the radical.

$x \sqrt[3]{4 z^{2}} \sqrt[3]{4 x^{5} z^{4}}$

Step 3

Rewrite $4 x^{5} z^{4}$ as ${(x z)}^{3} \cdot (4 x^{2} z)$ .

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Step 3.1

Factor out $x^{3}$ .

$x \sqrt[3]{4 z^{2}} \sqrt[3]{4 (x^{3} x^{2}) z^{4}}$

Step 3.2

Factor out $z^{3}$ .

$x \sqrt[3]{4 z^{2}} \sqrt[3]{4 (x^{3} x^{2}) (z^{3} z)}$

Step 3.3

Move $x^{2}$ .

$x \sqrt[3]{4 z^{2}} \sqrt[3]{4 (x^{3}) z^{3} x^{2} z}$

Step 3.4

Move $4$ .

$x \sqrt[3]{4 z^{2}} \sqrt[3]{(x^{3}) z^{3} \cdot 4 x^{2} z}$

Step 3.5

Rewrite $(x^{3}) z^{3}$ as ${(x z)}^{3}$ .

$x \sqrt[3]{4 z^{2}} \sqrt[3]{{(x z)}^{3} \cdot 4 x^{2} z}$

Step 3.6

Add parentheses.

$x \sqrt[3]{4 z^{2}} \sqrt[3]{{(x z)}^{3} \cdot 4 (x^{2} z)}$

Step 3.7

Add parentheses.

$x \sqrt[3]{4 z^{2}} \sqrt[3]{{(x z)}^{3} \cdot (4 x^{2} z)}$

Step 4

Pull terms out from under the radical.

$x \sqrt[3]{4 z^{2}} (x z \sqrt[3]{4 x^{2} z})$

Step 5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1

Move $x$ .

$x \cdot x \sqrt[3]{4 z^{2}} (z \sqrt[3]{4 x^{2} z})$

Step 5.2

Multiply $x$ by $x$ .

$x^{2} \sqrt[3]{4 z^{2}} (z \sqrt[3]{4 x^{2} z})$

Step 6

Multiply $x^{2} \sqrt[3]{4 z^{2}} (z \sqrt[3]{4 x^{2} z})$ .

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Step 6.1

Combine using the product rule for radicals.

$x^{2} (z \sqrt[3]{4 z^{2} (4 x^{2} z)})$

Step 6.2

Multiply $4$ by $4$ .

$x^{2} (z \sqrt[3]{16 z^{2} (x^{2} z)})$

Step 6.3

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

$x^{2} (z \sqrt[3]{16 (x^{2} (z^{2} z^{1}))})$

Step 6.4

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

$x^{2} (z \sqrt[3]{16 (x^{2} z^{2 + 1})})$

Step 6.5

Add $2$ and $1$ .

$x^{2} (z \sqrt[3]{16 (x^{2} z^{3})})$

Step 7

Rewrite $16 x^{2} z^{3}$ as ${(2 z)}^{3} \cdot (2 x^{2})$ .

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Step 7.1

Factor $8$ out of $16$ .

$x^{2} (z \sqrt[3]{8 (2) x^{2} z^{3}})$

Step 7.2

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

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

Step 7.3

Move $x^{2}$ .

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

Step 7.4

Move $2$ .

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

Step 7.5

Rewrite $2^{3} z^{3}$ as ${(2 z)}^{3}$ .

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

Step 7.6

Add parentheses.

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

Step 8

Pull terms out from under the radical.

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

Step 9

Simplify the expression.

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Step 9.1

Rewrite using the commutative property of multiplication.

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

Step 9.2

Multiply $z$ by $z$ .

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