Write the Fraction in Simplest Form ( cube root of 7)/( fifth root of 7)

$\frac{\sqrt[3]{7}}{\sqrt[5]{7}}$

Step 1

Multiply $\frac{\sqrt[3]{7}}{\sqrt[5]{7}}$ by $\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$\frac{\sqrt[3]{7}}{\sqrt[5]{7}} \cdot \frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{7}}{\sqrt[5]{7}}$ by $\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{\sqrt[5]{7} {\sqrt[5]{7}}^{4}}$

Raise $\sqrt[5]{7}$ to the power of $1$ .

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{1} {\sqrt[5]{7}}^{4}}$

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

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{5}}$

Rewrite ${\sqrt[5]{7}}^{5}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7}$ as $7^{\frac{1}{5}}$ .

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{{(7^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{1}{5} \cdot 5}}$

Combine $\frac{1}{5}$ and $5$ .

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{7^{1}}$

Evaluate the exponent.

$\frac{\sqrt[3]{7} {\sqrt[5]{7}}^{4}}{7}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[5]{7}}^{4}$ as ${(7^{4})}^{\frac{1}{5}}$ .

$\frac{\sqrt[3]{7} \sqrt[5]{7^{4}}}{7}$

Raise $7$ to the power of $4$ .

$\frac{\sqrt[3]{7} \sqrt[5]{2401}}{7}$

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $15$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{7^{\frac{1}{3}} \sqrt[5]{2401}}{7}$

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{5}{15}}$ .

$\frac{7^{\frac{5}{15}} \sqrt[5]{2401}}{7}$

Rewrite $7^{\frac{5}{15}}$ as $\sqrt[15]{7^{5}}$ .

$\frac{\sqrt[15]{7^{5}} \sqrt[5]{2401}}{7}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2401}$ as $2401^{\frac{1}{5}}$ .

$\frac{\sqrt[15]{7^{5}} \cdot 2401^{\frac{1}{5}}}{7}$

Rewrite $2401^{\frac{1}{5}}$ as $2401^{\frac{3}{15}}$ .

$\frac{\sqrt[15]{7^{5}} \cdot 2401^{\frac{3}{15}}}{7}$

Rewrite $2401^{\frac{3}{15}}$ as $\sqrt[15]{2401^{3}}$ .

$\frac{\sqrt[15]{7^{5}} \sqrt[15]{2401^{3}}}{7}$

Combine using the product rule for radicals.

$\frac{\sqrt[15]{7^{5} \cdot 2401^{3}}}{7}$

Rewrite $2401$ as $7^{4}$ .

$\frac{\sqrt[15]{7^{5} \cdot {(7^{4})}^{3}}}{7}$

Multiply the exponents in ${(7^{4})}^{3}$ .

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

$\frac{\sqrt[15]{7^{5} \cdot 7^{4 \cdot 3}}}{7}$

Multiply $4$ by $3$ .

$\frac{\sqrt[15]{7^{5} \cdot 7^{12}}}{7}$

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

$\frac{\sqrt[15]{7^{5 + 12}}}{7}$

Add $5$ and $12$ .

$\frac{\sqrt[15]{7^{17}}}{7}$

Step 5

Simplify the numerator.

Raise $7$ to the power of $17$ .

$\frac{\sqrt[15]{232630513987207}}{7}$

Rewrite $232630513987207$ as $7^{15} \cdot 49$ .

Factor $4747561509943$ out of $232630513987207$ .

$\frac{\sqrt[15]{4747561509943 (49)}}{7}$

Rewrite $4747561509943$ as $7^{15}$ .

$\frac{\sqrt[15]{7^{15} \cdot 49}}{7}$

Pull terms out from under the radical.

$\frac{7 \sqrt[15]{49}}{7}$

Step 6

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 \sqrt[15]{49}}{7}$

Divide $\sqrt[15]{49}$ by $1$ .

$\sqrt[15]{49}$

Write the Fraction in Simplest Form ( cube root of 7)/( fifth root of 7)

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