Precalculus Examples

$\frac{\sqrt[3]{8 x^{2}}}{\sqrt{x}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $8$ as $2^{3}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

Tap for more steps...

Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.6.2

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

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

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

Step 4.1.5

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

$\frac{2 (\sqrt[6]{x^{3}} x^{\frac{4}{6}})}{x}$

Step 4.1.6

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

$\frac{2 (\sqrt[6]{x^{3}} \sqrt[6]{x^{4}})}{x}$

Step 4.2

Combine using the product rule for radicals.

$\frac{2 \sqrt[6]{x^{3} x^{4}}}{x}$

Step 4.3

Multiply $x^{3}$ by $x^{4}$ 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.

$\frac{2 \sqrt[6]{x^{3 + 4}}}{x}$

Step 4.3.2

Add $3$ and $4$ .

$\frac{2 \sqrt[6]{x^{7}}}{x}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Factor out $x^{6}$ .

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

Step 5.2

Pull terms out from under the radical.

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

Step 6

Cancel the common factor of $x$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Divide $2 \sqrt[6]{x}$ by $1$ .

$2 \sqrt[6]{x}$