Simplify (4 square root of 3)/( cube root of 2)

$\frac{4 \sqrt{3}}{\sqrt[3]{2}}$

Step 1

Multiply $\frac{4 \sqrt{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

Multiply $\frac{4 \sqrt{3}}{\sqrt[3]{2}}$ by $\frac{{\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{2}}$ .

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

Raise $\sqrt[3]{2}$ to the power of $1$ .

$\frac{4 \sqrt{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1} {\sqrt[3]{2}}^{2}}$

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

$\frac{4 \sqrt{3} {\sqrt[3]{2}}^{2}}{{\sqrt[3]{2}}^{1 + 2}}$

Add $1$ and $2$ .

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

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

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

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{4 \sqrt{3} {\sqrt[3]{2}}^{2}}{2^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{4 \sqrt{3} {\sqrt[3]{2}}^{2}}{2^{1}}$

Evaluate the exponent.

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

Step 3

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \sqrt{3} {\sqrt[3]{2}}^{2}$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sqrt{3} {\sqrt[3]{2}}^{2}}{1}$

Divide $2 \sqrt{3} {\sqrt[3]{2}}^{2}$ by $1$ .

$2 \sqrt{3} {\sqrt[3]{2}}^{2}$

Step 4

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

$2 \sqrt{3} \sqrt[3]{2^{2}}$

Step 5

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

$2 \sqrt{3} \sqrt[3]{4}$

Step 6

Multiply $2 \sqrt{3} \sqrt[3]{4}$ .

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

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

$2 (4^{\frac{1}{3}} \sqrt{3})$

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

$2 \sqrt[6]{16 \cdot 3^{3}}$

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

$2 \sqrt[6]{16 \cdot 27}$

Multiply $16$ by $27$ .

$2 \sqrt[6]{432}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt[6]{432}$

Decimal Form:

$5.49891854 \dots$

Simplify (4 square root of 3)/( cube root of 2)

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