Precalculus Examples

${(\sqrt[3]{3} \cdot \sqrt{10})}^{4}$

Step 1

Multiply $\sqrt[3]{3} \sqrt{10}$ .

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Step 1.1

Rewrite the expression using the least common index of $6$ .

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Step 1.1.1

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

${(3^{\frac{1}{3}} \sqrt{10})}^{4}$

Step 1.1.2

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{2}{6}}$ .

${(3^{\frac{2}{6}} \sqrt{10})}^{4}$

Step 1.1.3

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

${(\sqrt[6]{3^{2}} \sqrt{10})}^{4}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

${(\sqrt[6]{3^{2}} \cdot 10^{\frac{1}{2}})}^{4}$

Step 1.1.5

Rewrite $10^{\frac{1}{2}}$ as $10^{\frac{3}{6}}$ .

${(\sqrt[6]{3^{2}} \cdot 10^{\frac{3}{6}})}^{4}$

Step 1.1.6

Rewrite $10^{\frac{3}{6}}$ as $\sqrt[6]{10^{3}}$ .

${(\sqrt[6]{3^{2}} \sqrt[6]{10^{3}})}^{4}$

Step 1.2

Combine using the product rule for radicals.

${\sqrt[6]{3^{2} \cdot 10^{3}}}^{4}$

Step 1.3

Raise $3$ to the power of $2$ .

${\sqrt[6]{9 \cdot 10^{3}}}^{4}$

Step 1.4

Raise $10$ to the power of $3$ .

${\sqrt[6]{9 \cdot 1000}}^{4}$

Step 1.5

Multiply $9$ by $1000$ .

${\sqrt[6]{9000}}^{4}$

Step 2

Rewrite ${\sqrt[6]{9000}}^{4}$ as $\sqrt[6]{9000^{4}}$ .

$\sqrt[6]{9000^{4}}$

Step 3

Raise $9000$ to the power of $4$ .

$\sqrt[6]{6561000000000000}$

Step 4

Rewrite $6561000000000000$ as $300^{6} \cdot 9$ .

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Step 4.1

Factor $729000000000000$ out of $6561000000000000$ .

$\sqrt[6]{729000000000000 (9)}$

Step 4.2

Rewrite $729000000000000$ as $300^{6}$ .

$\sqrt[6]{300^{6} \cdot 9}$

Step 5

Pull terms out from under the radical.

$300 \sqrt[6]{9}$

Step 6

Rewrite $9$ as $3^{2}$ .

$300 \sqrt[6]{3^{2}}$

Step 7

Rewrite $\sqrt[6]{3^{2}}$ as $\sqrt[3]{\sqrt{3^{2}}}$ .

$300 \sqrt[3]{\sqrt{3^{2}}}$

Step 8

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

$300 \sqrt[3]{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$300 \sqrt[3]{3}$

Decimal Form:

$432.67487109 \dots$