Simplify ( cube root of x^2)/( sixth root of x)

$\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x}}$

Step 1

Multiply $\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x}}$ by $\frac{{\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{5}}$ .

$\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x}} \cdot \frac{{\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{5}}$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{x^{2}}}{\sqrt[6]{x}}$ by $\frac{{\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{5}}$ .

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{\sqrt[6]{x} {\sqrt[6]{x}}^{5}}$

Raise $\sqrt[6]{x}$ to the power of $1$ .

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{1} {\sqrt[6]{x}}^{5}}$

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

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{1 + 5}}$

Add $1$ and $5$ .

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{6}}$

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

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

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{{(x^{\frac{1}{6}})}^{6}}$

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

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{x^{\frac{1}{6} \cdot 6}}$

Combine $\frac{1}{6}$ and $6$ .

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{x^{\frac{6}{6}}}$

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{x^{\frac{6}{6}}}$

Rewrite the expression.

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{x^{1}}$

Simplify.

$\frac{\sqrt[3]{x^{2}} {\sqrt[6]{x}}^{5}}{x}$

Step 3

Rewrite ${\sqrt[6]{x}}^{5}$ as $\sqrt[6]{x^{5}}$ .

$\frac{\sqrt[3]{x^{2}} \sqrt[6]{x^{5}}}{x}$

Step 4

Simplify the numerator.

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{x^{\frac{2}{3}} \sqrt[6]{x^{5}}}{x}$

Rewrite $x^{\frac{2}{3}}$ as $x^{\frac{4}{6}}$ .

$\frac{x^{\frac{4}{6}} \sqrt[6]{x^{5}}}{x}$

Rewrite $x^{\frac{4}{6}}$ as $\sqrt[6]{x^{4}}$ .

$\frac{\sqrt[6]{x^{4}} \sqrt[6]{x^{5}}}{x}$

Combine using the product rule for radicals.

$\frac{\sqrt[6]{x^{4} x^{5}}}{x}$

Multiply $x^{4}$ by $x^{5}$ by adding the exponents.

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

$\frac{\sqrt[6]{x^{4 + 5}}}{x}$

Add $4$ and $5$ .

$\frac{\sqrt[6]{x^{9}}}{x}$

Step 5

Simplify the numerator.

Factor out $x^{6}$ .

$\frac{\sqrt[6]{x^{6} x^{3}}}{x}$

Pull terms out from under the radical.

$\frac{x \sqrt[6]{x^{3}}}{x}$

Rewrite $\sqrt[6]{x^{3}}$ as $\sqrt{\sqrt[3]{x^{3}}}$ .

$\frac{x \sqrt{\sqrt[3]{x^{3}}}}{x}$

Pull terms out from under the radical, assuming real numbers.

$\frac{x \sqrt{x}}{x}$

Step 6

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x \sqrt{x}}{x}$

Divide $\sqrt{x}$ by $1$ .

$\sqrt{x}$

Simplify ( cube root of x^2)/( sixth root of x)

Related Questions

Support LCMGCF.com by disabling your adblocker.