Algebra Examples

$\frac{\sqrt[3]{18 y^{2}}}{\sqrt[3]{12 y}}$

Step 1

Combine $\sqrt[3]{18 y^{2}}$ and $\sqrt[3]{12 y}$ into a single radical.

$\sqrt[3]{\frac{18 y^{2}}{12 y}}$

Step 2

Reduce the expression $\frac{18 y^{2}}{12 y}$ by cancelling the common factors.

Tap for more steps...

Step 2.1

Factor $6$ out of $18 y^{2}$ .

$\sqrt[3]{\frac{6 (3 y^{2})}{12 y}}$

Step 2.2

Factor $6$ out of $12 y$ .

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

Step 2.3

Cancel the common factor.

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

Step 2.4

Rewrite the expression.

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

Step 3

Cancel the common factor of $y^{2}$ and $y$ .

Tap for more steps...

Step 3.1

Factor $y$ out of $3 y^{2}$ .

$\sqrt[3]{\frac{y (3 y)}{2 y}}$

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Factor $y$ out of $2 y$ .

$\sqrt[3]{\frac{y (3 y)}{y \cdot 2}}$

Step 3.2.2

Cancel the common factor.

$\sqrt[3]{\frac{y (3 y)}{y \cdot 2}}$

Step 3.2.3

Rewrite the expression.

$\sqrt[3]{\frac{3 y}{2}}$

Step 4

Rewrite $\sqrt[3]{\frac{3 y}{2}}$ as $\frac{\sqrt[3]{3 y}}{\sqrt[3]{2}}$ .

$\frac{\sqrt[3]{3 y}}{\sqrt[3]{2}}$

Step 5

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

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

Step 6

Combine and simplify the denominator.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 6.3

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

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

Step 6.4

Add $1$ and $2$ .

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

Step 6.5

Rewrite ${\sqrt[3]{2}}^{3}$ as $2$ .

Tap for more steps...

Step 6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

Step 6.5.2

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

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

Step 6.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 6.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.5.4.1

Cancel the common factor.

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

Step 6.5.4.2

Rewrite the expression.

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

Step 6.5.5

Evaluate the exponent.

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

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

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

Step 7.2

Raise $2$ to the power of $2$ .

$\frac{\sqrt[3]{3 y} \sqrt[3]{4}}{2}$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Combine using the product rule for radicals.

$\frac{\sqrt[3]{3 y \cdot 4}}{2}$

Step 8.2

Multiply $4$ by $3$ .

$\frac{\sqrt[3]{12 y}}{2}$