Simplify ( square root of 3)÷( cube root of 2)

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

Step 1

Rewrite the division as a fraction.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 4

Simplify the numerator.

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

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

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

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

Step 5

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

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

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

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

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

Step 6

Multiply $27$ by $16$ .

$\frac{\sqrt[6]{432}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[6]{432}}{2}$

Decimal Form:

$1.37472963 \dots$

Simplify ( square root of 3)÷( cube root of 2)

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