Finite Math Examples

$\sqrt[18]{467} \cdot \sqrt[11]{- 323}$

Step 1

Rewrite $- 323$ as ${(- 1)}^{11} \cdot 323$ .

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Step 1.1

Rewrite $- 323$ as $- 1 (323)$ .

$\sqrt[18]{467} \cdot \sqrt[11]{- 1 (323)}$

Step 1.2

Rewrite $- 1$ as ${(- 1)}^{11}$ .

$\sqrt[18]{467} \cdot \sqrt[11]{{(- 1)}^{11} \cdot 323}$

Step 2

Pull terms out from under the radical.

$\sqrt[18]{467} \cdot (- 1 \sqrt[11]{323})$

Step 3

Rewrite $- 1 \sqrt[11]{323}$ as $- \sqrt[11]{323}$ .

$\sqrt[18]{467} \cdot (- \sqrt[11]{323})$

Step 4

Multiply $\sqrt[18]{467} (- \sqrt[11]{323})$ .

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Step 4.1

Rewrite the expression using the least common index of $198$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[18]{467}$ as $467^{\frac{1}{18}}$ .

$- (467^{\frac{1}{18}} \sqrt[11]{323})$

Step 4.1.2

Rewrite $467^{\frac{1}{18}}$ as $467^{\frac{11}{198}}$ .

$- (467^{\frac{11}{198}} \sqrt[11]{323})$

Step 4.1.3

Rewrite $467^{\frac{11}{198}}$ as $\sqrt[198]{467^{11}}$ .

$- (\sqrt[198]{467^{11}} \sqrt[11]{323})$

Step 4.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{323}$ as $323^{\frac{1}{11}}$ .

$- (\sqrt[198]{467^{11}} \cdot 323^{\frac{1}{11}})$

Step 4.1.5

Rewrite $323^{\frac{1}{11}}$ as $323^{\frac{18}{198}}$ .

$- (\sqrt[198]{467^{11}} \cdot 323^{\frac{18}{198}})$

Step 4.1.6

Rewrite $323^{\frac{18}{198}}$ as $\sqrt[198]{323^{18}}$ .

$- (\sqrt[198]{467^{11}} \sqrt[198]{323^{18}})$

Step 4.2

Combine using the product rule for radicals.

$- \sqrt[198]{467^{11} \cdot 323^{18}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \sqrt[198]{467^{11} \cdot 323^{18}}$

Decimal Form:

$- 2.37905620 \dots$