Write the Fraction in Simplest Form ( sixth root of 3)/( cube root of 3)

$\frac{\sqrt[6]{3}}{\sqrt[3]{3}}$

Step 1

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

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

Step 2

Combine and simplify the denominator.

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 3

Simplify the numerator.

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

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

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

$\frac{\sqrt[6]{3} \sqrt[3]{9}}{3}$

Step 4

Simplify the numerator.

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$\frac{\sqrt[6]{3} \cdot 9^{\frac{1}{3}}}{3}$

Rewrite $9^{\frac{1}{3}}$ as $9^{\frac{2}{6}}$ .

$\frac{\sqrt[6]{3} \cdot 9^{\frac{2}{6}}}{3}$

Rewrite $9^{\frac{2}{6}}$ as $\sqrt[6]{9^{2}}$ .

$\frac{\sqrt[6]{3} \sqrt[6]{9^{2}}}{3}$

Combine using the product rule for radicals.

$\frac{\sqrt[6]{3 \cdot 9^{2}}}{3}$

Rewrite $9$ as $3^{2}$ .

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

Multiply the exponents in ${(3^{2})}^{2}$ .

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

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

Multiply $2$ by $2$ .

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

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

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

Add $1$ and $4$ .

$\frac{\sqrt[6]{3^{5}}}{3}$

Step 5

Raise $3$ to the power of $5$ .

$\frac{\sqrt[6]{243}}{3}$

Write the Fraction in Simplest Form ( sixth root of 3)/( cube root of 3)

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