Finite Math Examples

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

Step 1

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

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

Step 2

Pull terms out from under the radical.

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

Step 3

Multiply $x \sqrt[3]{4} \sqrt[4]{5}$ .

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Step 3.1

Rewrite the expression using the least common index of $12$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$x (5^{\frac{1}{4}} \sqrt[3]{4})$

Step 3.1.2

Rewrite $5^{\frac{1}{4}}$ as $5^{\frac{3}{12}}$ .

$x (5^{\frac{3}{12}} \sqrt[3]{4})$

Step 3.1.3

Rewrite $5^{\frac{3}{12}}$ as $\sqrt[12]{5^{3}}$ .

$x (\sqrt[12]{5^{3}} \sqrt[3]{4})$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{4}$ as $4^{\frac{1}{3}}$ .

$x (\sqrt[12]{5^{3}} \cdot 4^{\frac{1}{3}})$

Step 3.1.5

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{4}{12}}$ .

$x (\sqrt[12]{5^{3}} \cdot 4^{\frac{4}{12}})$

Step 3.1.6

Rewrite $4^{\frac{4}{12}}$ as $\sqrt[12]{4^{4}}$ .

$x (\sqrt[12]{5^{3}} \sqrt[12]{4^{4}})$

Step 3.2

Combine using the product rule for radicals.

$x \sqrt[12]{5^{3} \cdot 4^{4}}$

Step 3.3

Raise $5$ to the power of $3$ .

$x \sqrt[12]{125 \cdot 4^{4}}$

Step 3.4

Raise $4$ to the power of $4$ .

$x \sqrt[12]{125 \cdot 256}$

Step 3.5

Multiply $125$ by $256$ .

$x \sqrt[12]{32000}$