Write the Fraction in Simplest Form (10 root of 7)/( fifth root of 7)

$\frac{\sqrt[10]{7}}{\sqrt[5]{7}}$

Step 1

Multiply $\frac{\sqrt[10]{7}}{\sqrt[5]{7}}$ by $\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$\frac{\sqrt[10]{7}}{\sqrt[5]{7}} \cdot \frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[10]{7}}{\sqrt[5]{7}}$ by $\frac{{\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{4}}$ .

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{\sqrt[5]{7} {\sqrt[5]{7}}^{4}}$

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

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{1} {\sqrt[5]{7}}^{4}}$

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

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{1 + 4}}$

Add $1$ and $4$ .

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{{\sqrt[5]{7}}^{5}}$

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

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

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{{(7^{\frac{1}{5}})}^{5}}$

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

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{1}{5} \cdot 5}}$

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

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}}$

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{7^{\frac{5}{5}}}$

Rewrite the expression.

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{7^{1}}$

Evaluate the exponent.

$\frac{\sqrt[10]{7} {\sqrt[5]{7}}^{4}}{7}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[5]{7}}^{4}$ as $\sqrt[5]{7^{4}}$ .

$\frac{\sqrt[10]{7} \sqrt[5]{7^{4}}}{7}$

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

$\frac{\sqrt[10]{7} \sqrt[5]{2401}}{7}$

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $10$ .

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

$\frac{\sqrt[10]{7} \cdot 2401^{\frac{1}{5}}}{7}$

Rewrite $2401^{\frac{1}{5}}$ as $2401^{\frac{2}{10}}$ .

$\frac{\sqrt[10]{7} \cdot 2401^{\frac{2}{10}}}{7}$

Rewrite $2401^{\frac{2}{10}}$ as $\sqrt[10]{2401^{2}}$ .

$\frac{\sqrt[10]{7} \sqrt[10]{2401^{2}}}{7}$

Combine using the product rule for radicals.

$\frac{\sqrt[10]{7 \cdot 2401^{2}}}{7}$

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

$\frac{\sqrt[10]{7 \cdot {(7^{4})}^{2}}}{7}$

Multiply the exponents in ${(7^{4})}^{2}$ .

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

$\frac{\sqrt[10]{7 \cdot 7^{4 \cdot 2}}}{7}$

Multiply $4$ by $2$ .

$\frac{\sqrt[10]{7 \cdot 7^{8}}}{7}$

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

$\frac{\sqrt[10]{7^{1 + 8}}}{7}$

Add $1$ and $8$ .

$\frac{\sqrt[10]{7^{9}}}{7}$

Step 5

Raise $7$ to the power of $9$ .

$\frac{\sqrt[10]{40353607}}{7}$

Write the Fraction in Simplest Form (10 root of 7)/( fifth root of 7)

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