Simplify square root of 3x^2 cube root of 6x^3 fourth root of 4x^2

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

Reorder $3$ and $x^{2}$ .

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

Pull terms out from under the radical.

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

Reorder $6$ and $x^{3}$ .

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

Pull terms out from under the radical.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

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

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

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

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

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

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

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

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

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

Combine using the product rule for radicals.

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

Raise $6$ to the power of $2$ .

$x^{2} \sqrt[6]{36 \cdot 3^{3}} \sqrt[4]{4 x^{2}}$

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

$x^{2} \sqrt[6]{36 \cdot 27} \sqrt[4]{4 x^{2}}$

Multiply $36$ by $27$ .

$x^{2} \sqrt[6]{972} \sqrt[4]{4 x^{2}}$

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$x^{2} \sqrt[6]{972} \sqrt[4]{{(2 x)}^{2}}$

Rewrite $\sqrt[4]{{(2 x)}^{2}}$ as $\sqrt{\sqrt{{(2 x)}^{2}}}$ .

$x^{2} \sqrt[6]{972} \sqrt{\sqrt{{(2 x)}^{2}}}$

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

$x^{2} \sqrt[6]{972} \sqrt{2 x}$

Multiply $x^{2} \sqrt[6]{972} \sqrt{2 x}$ .

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

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

$x^{2} ({(2 x)}^{\frac{1}{2}} \sqrt[6]{972})$

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

$x^{2} ({(2 x)}^{\frac{3}{6}} \sqrt[6]{972})$

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

$x^{2} (\sqrt[6]{{(2 x)}^{3}} \sqrt[6]{972})$

Combine using the product rule for radicals.

$x^{2} \sqrt[6]{{(2 x)}^{3} \cdot 972}$

Apply the product rule to $2 x$ .

$x^{2} \sqrt[6]{2^{3} x^{3} \cdot 972}$

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

$x^{2} \sqrt[6]{8 x^{3} \cdot 972}$

Multiply $972$ by $8$ .

$x^{2} \sqrt[6]{7776 x^{3}}$

Simplify square root of 3x^2 cube root of 6x^3 fourth root of 4x^2

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