Precalculus Examples

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x}}{\sqrt[6]{x}}$

Step 1

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

$\sqrt[4]{x} \cdot (\frac{\sqrt[3]{x}}{\sqrt[6]{x}} \cdot \frac{{\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{5}})$

Step 2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{\sqrt[6]{x} {\sqrt[6]{x}}^{5}}$

Step 2.2

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{1} {\sqrt[6]{x}}^{5}}$

Step 2.3

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{1 + 5}}$

Step 2.4

Add $1$ and $5$ .

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{{\sqrt[6]{x}}^{6}}$

Step 2.5

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

Tap for more steps...

Step 2.5.1

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{{(x^{\frac{1}{6}})}^{6}}$

Step 2.5.2

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{x^{\frac{1}{6} \cdot 6}}$

Step 2.5.3

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{x^{\frac{6}{6}}}$

Step 2.5.4

Cancel the common factor of $6$ .

Tap for more steps...

Step 2.5.4.1

Cancel the common factor.

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{x^{\frac{6}{6}}}$

Step 2.5.4.2

Rewrite the expression.

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{x^{1}}$

Step 2.5.5

Simplify.

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} {\sqrt[6]{x}}^{5}}{x}$

Step 3

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

$\sqrt[4]{x} \cdot \frac{\sqrt[3]{x} \sqrt[6]{x^{5}}}{x}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

$\sqrt[4]{x} \cdot \frac{x^{\frac{1}{3}} \sqrt[6]{x^{5}}}{x}$

Step 4.1.2

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$\sqrt[4]{x} \cdot \frac{x^{\frac{2}{6}} \sqrt[6]{x^{5}}}{x}$

Step 4.1.3

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$\sqrt[4]{x} \cdot \frac{\sqrt[6]{x^{2}} \sqrt[6]{x^{5}}}{x}$

Step 4.2

Combine using the product rule for radicals.

$\sqrt[4]{x} \cdot \frac{\sqrt[6]{x^{2} x^{5}}}{x}$

Step 4.3

Multiply $x^{2}$ by $x^{5}$ by adding the exponents.

Tap for more steps...

Step 4.3.1

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

$\sqrt[4]{x} \cdot \frac{\sqrt[6]{x^{2 + 5}}}{x}$

Step 4.3.2

Add $2$ and $5$ .

$\sqrt[4]{x} \cdot \frac{\sqrt[6]{x^{7}}}{x}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Factor out $x^{6}$ .

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

Step 5.2

Pull terms out from under the radical.

$\sqrt[4]{x} \cdot \frac{x \sqrt[6]{x}}{x}$

Step 6

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\sqrt[4]{x} \cdot \frac{x \sqrt[6]{x}}{x}$

Step 6.2

Divide $\sqrt[6]{x}$ by $1$ .

$\sqrt[4]{x} \cdot \sqrt[6]{x}$

Step 7

Multiply $\sqrt[4]{x} \sqrt[6]{x}$ .

Tap for more steps...

Step 7.1

Rewrite the expression using the least common index of $12$ .

Tap for more steps...

Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$x^{\frac{1}{4}} \sqrt[6]{x}$

Step 7.1.2

Rewrite $x^{\frac{1}{4}}$ as $x^{\frac{3}{12}}$ .

$x^{\frac{3}{12}} \sqrt[6]{x}$

Step 7.1.3

Rewrite $x^{\frac{3}{12}}$ as $\sqrt[12]{x^{3}}$ .

$\sqrt[12]{x^{3}} \sqrt[6]{x}$

Step 7.1.4

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

$\sqrt[12]{x^{3}} x^{\frac{1}{6}}$

Step 7.1.5

Rewrite $x^{\frac{1}{6}}$ as $x^{\frac{2}{12}}$ .

$\sqrt[12]{x^{3}} x^{\frac{2}{12}}$

Step 7.1.6

Rewrite $x^{\frac{2}{12}}$ as $\sqrt[12]{x^{2}}$ .

$\sqrt[12]{x^{3}} \sqrt[12]{x^{2}}$

Step 7.2

Combine using the product rule for radicals.

$\sqrt[12]{x^{3} x^{2}}$

Step 7.3

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 7.3.1

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

$\sqrt[12]{x^{3 + 2}}$

Step 7.3.2

Add $3$ and $2$ .

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