Write the Fraction in Simplest Form ( cube root of 5)/( fourth root of 5)

$\frac{\sqrt[3]{5}}{\sqrt[4]{5}}$

Step 1

Multiply $\frac{\sqrt[3]{5}}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$\frac{\sqrt[3]{5}}{\sqrt[4]{5}} \cdot \frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{5}}{\sqrt[4]{5}}$ by $\frac{{\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{3}}$ .

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{\sqrt[4]{5} {\sqrt[4]{5}}^{3}}$

Raise $\sqrt[4]{5}$ to the power of $1$ .

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1} {\sqrt[4]{5}}^{3}}$

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

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{1 + 3}}$

Add $1$ and $3$ .

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{{\sqrt[4]{5}}^{4}}$

Rewrite ${\sqrt[4]{5}}^{4}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{5}$ as $5^{\frac{1}{4}}$ .

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{{(5^{\frac{1}{4}})}^{4}}$

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

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{5^{\frac{1}{4} \cdot 4}}$

Combine $\frac{1}{4}$ and $4$ .

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{5^{\frac{4}{4}}}$

Rewrite the expression.

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{5^{1}}$

Evaluate the exponent.

$\frac{\sqrt[3]{5} {\sqrt[4]{5}}^{3}}{5}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[4]{5}}^{3}$ as ${(5^{3})}^{\frac{1}{4}}$ .

$\frac{\sqrt[3]{5} \sqrt[4]{5^{3}}}{5}$

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

$\frac{\sqrt[3]{5} \sqrt[4]{125}}{5}$

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $12$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$\frac{5^{\frac{1}{3}} \sqrt[4]{125}}{5}$

Rewrite $5^{\frac{1}{3}}$ as $5^{\frac{4}{12}}$ .

$\frac{5^{\frac{4}{12}} \sqrt[4]{125}}{5}$

Rewrite $5^{\frac{4}{12}}$ as $\sqrt[12]{5^{4}}$ .

$\frac{\sqrt[12]{5^{4}} \sqrt[4]{125}}{5}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{125}$ as $125^{\frac{1}{4}}$ .

$\frac{\sqrt[12]{5^{4}} \cdot 125^{\frac{1}{4}}}{5}$

Rewrite $125^{\frac{1}{4}}$ as $125^{\frac{3}{12}}$ .

$\frac{\sqrt[12]{5^{4}} \cdot 125^{\frac{3}{12}}}{5}$

Rewrite $125^{\frac{3}{12}}$ as $\sqrt[12]{125^{3}}$ .

$\frac{\sqrt[12]{5^{4}} \sqrt[12]{125^{3}}}{5}$

Combine using the product rule for radicals.

$\frac{\sqrt[12]{5^{4} \cdot 125^{3}}}{5}$

Rewrite $125$ as $5^{3}$ .

$\frac{\sqrt[12]{5^{4} \cdot {(5^{3})}^{3}}}{5}$

Multiply the exponents in ${(5^{3})}^{3}$ .

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

$\frac{\sqrt[12]{5^{4} \cdot 5^{3 \cdot 3}}}{5}$

Multiply $3$ by $3$ .

$\frac{\sqrt[12]{5^{4} \cdot 5^{9}}}{5}$

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

$\frac{\sqrt[12]{5^{4 + 9}}}{5}$

Add $4$ and $9$ .

$\frac{\sqrt[12]{5^{13}}}{5}$

Step 5

Simplify the numerator.

Raise $5$ to the power of $13$ .

$\frac{\sqrt[12]{1220703125}}{5}$

Rewrite $1220703125$ as $5^{12} \cdot 5$ .

Factor $244140625$ out of $1220703125$ .

$\frac{\sqrt[12]{244140625 (5)}}{5}$

Rewrite $244140625$ as $5^{12}$ .

$\frac{\sqrt[12]{5^{12} \cdot 5}}{5}$

Pull terms out from under the radical.

$\frac{5 \sqrt[12]{5}}{5}$

Step 6

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 \sqrt[12]{5}}{5}$

Divide $\sqrt[12]{5}$ by $1$ .

$\sqrt[12]{5}$

Write the Fraction in Simplest Form ( cube root of 5)/( fourth root of 5)

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