Algebra Examples

$\sqrt{2 x} \cdot \sqrt[6]{3 x}$

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 $\sqrt{2 x}$ as ${(2 x)}^{\frac{1}{2}}$ .

${(2 x)}^{\frac{1}{2}} \sqrt[6]{3 x}$

Step 1.2

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

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

Step 1.3

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

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

Step 2

Combine using the product rule for radicals.

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

Step 3

Apply the product rule to $2 x$ .

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

Step 4

Combine exponents.

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Step 4.1

Raise $x$ to the power of $1$ .

$\sqrt[6]{2^{3} \cdot 3 (x^{3} x^{1})}$

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt[6]{2^{3} \cdot 3 x^{3 + 1}}$

Step 4.3

Add $3$ and $1$ .

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

Step 5

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

$\sqrt[6]{8 \cdot 3 x^{4}}$

Step 6

Multiply $8$ by $3$ .

$\sqrt[6]{24 x^{4}}$