Precalculus Examples

$\sqrt[3]{2 a} \cdot \sqrt{b}$

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 $\sqrt[3]{2 a}$ as ${(2 a)}^{\frac{1}{3}}$ .

${(2 a)}^{\frac{1}{3}} \sqrt{b}$

Step 1.2

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

${(2 a)}^{\frac{2}{6}} \sqrt{b}$

Step 1.3

Rewrite ${(2 a)}^{\frac{2}{6}}$ as $\sqrt[6]{{(2 a)}^{2}}$ .

$\sqrt[6]{{(2 a)}^{2}} \sqrt{b}$

Step 1.4

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

$\sqrt[6]{{(2 a)}^{2}} b^{\frac{1}{2}}$

Step 1.5

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

$\sqrt[6]{{(2 a)}^{2}} b^{\frac{3}{6}}$

Step 1.6

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

$\sqrt[6]{{(2 a)}^{2}} \sqrt[6]{b^{3}}$

Step 2

Combine using the product rule for radicals.

$\sqrt[6]{{(2 a)}^{2} b^{3}}$

Step 3

Apply the product rule to $2 a$ .

$\sqrt[6]{2^{2} a^{2} b^{3}}$

Step 4

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

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