Basic Math Examples

$\frac{3 \sqrt{243 y^{8}}}{\sqrt[3]{9}}$

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $243 y^{8}$ as ${(9 y^{4})}^{2} \cdot 3$ .

Tap for more steps...

Step 1.1.1

Factor $81$ out of $243$ .

$\frac{3 \sqrt{81 (3) y^{8}}}{\sqrt[3]{9}}$

Step 1.1.2

Rewrite $81$ as $9^{2}$ .

$\frac{3 \sqrt{9^{2} \cdot 3 y^{8}}}{\sqrt[3]{9}}$

Step 1.1.3

Rewrite $y^{8}$ as ${(y^{4})}^{2}$ .

$\frac{3 \sqrt{9^{2} \cdot 3 {(y^{4})}^{2}}}{\sqrt[3]{9}}$

Step 1.1.4

Move $3$ .

$\frac{3 \sqrt{9^{2} {(y^{4})}^{2} \cdot 3}}{\sqrt[3]{9}}$

Step 1.1.5

Rewrite $9^{2} {(y^{4})}^{2}$ as ${(9 y^{4})}^{2}$ .

$\frac{3 \sqrt{{(9 y^{4})}^{2} \cdot 3}}{\sqrt[3]{9}}$

Step 1.2

Pull terms out from under the radical.

$\frac{3 \cdot 9 y^{4} \sqrt{3}}{\sqrt[3]{9}}$

Step 1.3

Multiply $3$ by $9$ .

$\frac{27 y^{4} \sqrt{3}}{\sqrt[3]{9}}$

Step 2

Multiply $\frac{27 y^{4} \sqrt{3}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{27 y^{4} \sqrt{3}}{\sqrt[3]{9}} \cdot \frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{27 y^{4} \sqrt{3}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Step 3.2

Raise $\sqrt[3]{9}$ to the power of $1$ .

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

Step 3.3

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

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Step 3.4

Add $1$ and $2$ .

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Step 3.5

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

Tap for more steps...

Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{{(9^{\frac{1}{3}})}^{3}}$

Step 3.5.2

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

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Step 3.5.3

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

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.5.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.4.1

Cancel the common factor.

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Step 3.5.4.2

Rewrite the expression.

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{9^{1}}$

Step 3.5.5

Evaluate the exponent.

$\frac{27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{9}$

Step 4

Cancel the common factor of $27$ and $9$ .

Tap for more steps...

Step 4.1

Factor $9$ out of $27 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}$ .

$\frac{9 (3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2})}{9}$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $9$ out of $9$ .

$\frac{9 (3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2})}{9 (1)}$

Step 4.2.2

Cancel the common factor.

$\frac{9 (3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2})}{9 \cdot 1}$

Step 4.2.3

Rewrite the expression.

$\frac{3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}}{1}$

Step 4.2.4

Divide $3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}$ by $1$ .

$3 y^{4} \sqrt{3} {\sqrt[3]{9}}^{2}$

Step 5

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$3 y^{4} \sqrt{3} \sqrt[3]{9^{2}}$

Step 6

Raise $9$ to the power of $2$ .

$3 y^{4} \sqrt{3} \sqrt[3]{81}$

Step 7

Rewrite $81$ as $3^{3} \cdot 3$ .

Tap for more steps...

Step 7.1

Factor $27$ out of $81$ .

$3 y^{4} \sqrt{3} \sqrt[3]{27 (3)}$

Step 7.2

Rewrite $27$ as $3^{3}$ .

$3 y^{4} \sqrt{3} \sqrt[3]{3^{3} \cdot 3}$

Step 8

Pull terms out from under the radical.

$3 y^{4} \sqrt{3} (3 \sqrt[3]{3})$

Step 9

Multiply $3 y^{4} \sqrt{3} (3 \sqrt[3]{3})$ .

Tap for more steps...

Step 9.1

Multiply $3$ by $3$ .

$9 y^{4} \sqrt{3} \sqrt[3]{3}$

Step 9.2

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

Tap for more steps...

Step 9.2.1

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

$9 y^{4} (3^{\frac{1}{3}} \sqrt{3})$

Step 9.2.2

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

$9 y^{4} (3^{\frac{2}{6}} \sqrt{3})$

Step 9.2.3

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

$9 y^{4} (\sqrt[6]{3^{2}} \sqrt{3})$

Step 9.2.4

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

$9 y^{4} (\sqrt[6]{3^{2}} \cdot 3^{\frac{1}{2}})$

Step 9.2.5

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

$9 y^{4} (\sqrt[6]{3^{2}} \cdot 3^{\frac{3}{6}})$

Step 9.2.6

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

$9 y^{4} (\sqrt[6]{3^{2}} \sqrt[6]{3^{3}})$

Step 9.3

Combine using the product rule for radicals.

$9 y^{4} \sqrt[6]{3^{2} \cdot 3^{3}}$

Step 9.4

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

Tap for more steps...

Step 9.4.1

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

$9 y^{4} \sqrt[6]{3^{2 + 3}}$

Step 9.4.2

Add $2$ and $3$ .

$9 y^{4} \sqrt[6]{3^{5}}$

Step 10

Raise $3$ to the power of $5$ .

$9 y^{4} \sqrt[6]{243}$