Simplify sixth root of x* fifth root of x*12 root of x^5

$\sqrt[6]{x} \cdot \sqrt[5]{x} \cdot \sqrt[12]{x^{5}}$

Multiply $\sqrt[6]{x} \sqrt[5]{x}$ .

Rewrite the expression using the least common index of $30$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$x^{\frac{1}{6}} \sqrt[5]{x} \cdot \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{1}{6}}$ as $x^{\frac{5}{30}}$ .

$x^{\frac{5}{30}} \sqrt[5]{x} \cdot \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{5}{30}}$ as $\sqrt[30]{x^{5}}$ .

$\sqrt[30]{x^{5}} \sqrt[5]{x} \cdot \sqrt[12]{x^{5}}$

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

$\sqrt[30]{x^{5}} x^{\frac{1}{5}} \cdot \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{1}{5}}$ as $x^{\frac{6}{30}}$ .

$\sqrt[30]{x^{5}} x^{\frac{6}{30}} \cdot \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{6}{30}}$ as $\sqrt[30]{x^{6}}$ .

$\sqrt[30]{x^{5}} \sqrt[30]{x^{6}} \cdot \sqrt[12]{x^{5}}$

Combine using the product rule for radicals.

$\sqrt[30]{x^{5} x^{6}} \cdot \sqrt[12]{x^{5}}$

Multiply $x^{5}$ by $x^{6}$ by adding the exponents.

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

$\sqrt[30]{x^{5 + 6}} \cdot \sqrt[12]{x^{5}}$

Add $5$ and $6$ .

$\sqrt[30]{x^{11}} \cdot \sqrt[12]{x^{5}}$

Multiply $\sqrt[30]{x^{11}} \sqrt[12]{x^{5}}$ .

Rewrite the expression using the least common index of $60$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[30]{x^{11}}$ as $x^{\frac{11}{30}}$ .

$x^{\frac{11}{30}} \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{11}{30}}$ as $x^{\frac{22}{60}}$ .

$x^{\frac{22}{60}} \sqrt[12]{x^{5}}$

Rewrite $x^{\frac{22}{60}}$ as $\sqrt[60]{x^{22}}$ .

$\sqrt[60]{x^{22}} \sqrt[12]{x^{5}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{x^{5}}$ as $x^{\frac{5}{12}}$ .

$\sqrt[60]{x^{22}} x^{\frac{5}{12}}$

Rewrite $x^{\frac{5}{12}}$ as $x^{\frac{25}{60}}$ .

$\sqrt[60]{x^{22}} x^{\frac{25}{60}}$

Rewrite $x^{\frac{25}{60}}$ as $\sqrt[60]{x^{25}}$ .

$\sqrt[60]{x^{22}} \sqrt[60]{x^{25}}$

Combine using the product rule for radicals.

$\sqrt[60]{x^{22} x^{25}}$

Multiply $x^{22}$ by $x^{25}$ by adding the exponents.

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

$\sqrt[60]{x^{22 + 25}}$

Add $22$ and $25$ .

$\sqrt[60]{x^{47}}$