Trigonometry Examples

$\sqrt[3]{4} \sqrt{7}$

Step 1

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

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Step 1.1

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

$4^{\frac{1}{3}} \sqrt{7}$

Step 1.2

Rewrite $4^{\frac{1}{3}}$ as $4^{\frac{2}{6}}$ .

$4^{\frac{2}{6}} \sqrt{7}$

Step 1.3

Rewrite $4^{\frac{2}{6}}$ as $\sqrt[6]{4^{2}}$ .

$\sqrt[6]{4^{2}} \sqrt{7}$

Step 1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt[6]{4^{2}} \cdot 7^{\frac{1}{2}}$

Step 1.5

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{3}{6}}$ .

$\sqrt[6]{4^{2}} \cdot 7^{\frac{3}{6}}$

Step 1.6

Rewrite $7^{\frac{3}{6}}$ as $\sqrt[6]{7^{3}}$ .

$\sqrt[6]{4^{2}} \sqrt[6]{7^{3}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[6]{4^{2} \cdot 7^{3}}$

Step 3

Raise $4$ to the power of $2$ .

$\sqrt[6]{16 \cdot 7^{3}}$

Step 4

Raise $7$ to the power of $3$ .

$\sqrt[6]{16 \cdot 343}$

Step 5

Multiply $16$ by $343$ .

$\sqrt[6]{5488}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{5488}$

Decimal Form:

$4.19986841 \dots$