Simplify ( seventh root of x^2)( fifth root of x^3)

$(\sqrt[7]{x^{2}}) (\sqrt[5]{x^{3}})$

Rewrite the expression using the least common index of $35$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{x^{2}}$ as $x^{\frac{2}{7}}$ .

$x^{\frac{2}{7}} \sqrt[5]{x^{3}}$

Rewrite $x^{\frac{2}{7}}$ as $x^{\frac{10}{35}}$ .

$x^{\frac{10}{35}} \sqrt[5]{x^{3}}$

Rewrite $x^{\frac{10}{35}}$ as $\sqrt[35]{x^{10}}$ .

$\sqrt[35]{x^{10}} \sqrt[5]{x^{3}}$

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

$\sqrt[35]{x^{10}} x^{\frac{3}{5}}$

Rewrite $x^{\frac{3}{5}}$ as $x^{\frac{21}{35}}$ .

$\sqrt[35]{x^{10}} x^{\frac{21}{35}}$

Rewrite $x^{\frac{21}{35}}$ as $\sqrt[35]{x^{21}}$ .

$\sqrt[35]{x^{10}} \sqrt[35]{x^{21}}$

Combine using the product rule for radicals.

$\sqrt[35]{x^{10} x^{21}}$

Multiply $x^{10}$ by $x^{21}$ by adding the exponents.

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

$\sqrt[35]{x^{10 + 21}}$

Add $10$ and $21$ .

$\sqrt[35]{x^{31}}$