Basic Math Examples

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Rewrite $3 m^{4}$ as $m^{3} \cdot (3 m)$ .

Tap for more steps...

Step 1.1.1

Factor out $m^{3}$ .

$\frac{\sqrt[3]{3 (m^{3} m)}}{\sqrt[9]{27 m^{2}}}$

Step 1.1.2

Reorder $3$ and $m^{3}$ .

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

Step 1.1.3

Add parentheses.

$\frac{\sqrt[3]{m^{3} \cdot (3 m)}}{\sqrt[9]{27 m^{2}}}$

Step 1.2

Pull terms out from under the radical.

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

Step 2

Multiply $\frac{m \sqrt[3]{3 m}}{\sqrt[9]{27 m^{2}}}$ by $\frac{{\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{8}}$ .

$\frac{m \sqrt[3]{3 m}}{\sqrt[9]{27 m^{2}}} \cdot \frac{{\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{8}}$

Step 3

Simplify terms.

Tap for more steps...

Step 3.1

Combine and simplify the denominator.

Tap for more steps...

Step 3.1.1

Multiply $\frac{m \sqrt[3]{3 m}}{\sqrt[9]{27 m^{2}}}$ by $\frac{{\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{8}}$ .

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{\sqrt[9]{27 m^{2}} {\sqrt[9]{27 m^{2}}}^{8}}$

Step 3.1.2

Raise $\sqrt[9]{27 m^{2}}$ to the power of $1$ .

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{1} {\sqrt[9]{27 m^{2}}}^{8}}$

Step 3.1.3

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

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{1 + 8}}$

Step 3.1.4

Add $1$ and $8$ .

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{\sqrt[9]{27 m^{2}}}^{9}}$

Step 3.1.5

Rewrite ${\sqrt[9]{27 m^{2}}}^{9}$ as $27 m^{2}$ .

Tap for more steps...

Step 3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{27 m^{2}}$ as ${(27 m^{2})}^{\frac{1}{9}}$ .

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{({(27 m^{2})}^{\frac{1}{9}})}^{9}}$

Step 3.1.5.2

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

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{(27 m^{2})}^{\frac{1}{9} \cdot 9}}$

Step 3.1.5.3

Combine $\frac{1}{9}$ and $9$ .

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{(27 m^{2})}^{\frac{9}{9}}}$

Step 3.1.5.4

Cancel the common factor of $9$ .

Tap for more steps...

Step 3.1.5.4.1

Cancel the common factor.

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{(27 m^{2})}^{\frac{9}{9}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{{(27 m^{2})}^{1}}$

Step 3.1.5.5

Simplify.

$\frac{m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{27 m^{2}}$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Factor $m$ out of $m \sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}$ .

$\frac{m (\sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8})}{27 m^{2}}$

Step 3.2.2

Cancel the common factors.

Tap for more steps...

Step 3.2.2.1

Factor $m$ out of $27 m^{2}$ .

$\frac{m (\sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8})}{m (27 m)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{m (\sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8})}{m (27 m)}$

Step 3.2.2.3

Rewrite the expression.

$\frac{\sqrt[3]{3 m} {\sqrt[9]{27 m^{2}}}^{8}}{27 m}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Rewrite ${\sqrt[9]{27 m^{2}}}^{8}$ as $\sqrt[9]{{(27 m^{2})}^{8}}$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{{(27 m^{2})}^{8}}}{27 m}$

Step 4.2

Apply the product rule to $27 m^{2}$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{27^{8} {(m^{2})}^{8}}}{27 m}$

Step 4.3

Raise $27$ to the power of $8$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{282429536481 {(m^{2})}^{8}}}{27 m}$

Step 4.4

Multiply the exponents in ${(m^{2})}^{8}$ .

Tap for more steps...

Step 4.4.1

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

$\frac{\sqrt[3]{3 m} \sqrt[9]{282429536481 m^{2 \cdot 8}}}{27 m}$

Step 4.4.2

Multiply $2$ by $8$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{282429536481 m^{16}}}{27 m}$

Step 4.5

Rewrite $282429536481 m^{16}$ as ${(9 m)}^{9} \cdot (729 m^{7})$ .

Tap for more steps...

Step 4.5.1

Factor $387420489$ out of $282429536481$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{387420489 (729) m^{16}}}{27 m}$

Step 4.5.2

Rewrite $387420489$ as $9^{9}$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{9^{9} \cdot 729 m^{16}}}{27 m}$

Step 4.5.3

Factor out $m^{9}$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{9^{9} \cdot 729 (m^{9} m^{7})}}{27 m}$

Step 4.5.4

Move $729$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{9^{9} m^{9} \cdot 729 m^{7}}}{27 m}$

Step 4.5.5

Rewrite $9^{9} m^{9}$ as ${(9 m)}^{9}$ .

$\frac{\sqrt[3]{3 m} \sqrt[9]{{(9 m)}^{9} \cdot 729 m^{7}}}{27 m}$

Step 4.5.6

Add parentheses.

$\frac{\sqrt[3]{3 m} \sqrt[9]{{(9 m)}^{9} \cdot (729 m^{7})}}{27 m}$

Step 4.6

Pull terms out from under the radical.

$\frac{\sqrt[3]{3 m} \cdot 9 m \sqrt[9]{729 m^{7}}}{27 m}$

Step 4.7

Combine exponents.

Tap for more steps...

Step 4.7.1

Rewrite the expression using the least common index of $9$ .

Tap for more steps...

Step 4.7.1.1

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

$\frac{9 m ({(3 m)}^{\frac{1}{3}} \sqrt[9]{729 m^{7}})}{27 m}$

Step 4.7.1.2

Rewrite ${(3 m)}^{\frac{1}{3}}$ as ${(3 m)}^{\frac{3}{9}}$ .

$\frac{9 m ({(3 m)}^{\frac{3}{9}} \sqrt[9]{729 m^{7}})}{27 m}$

Step 4.7.1.3

Rewrite ${(3 m)}^{\frac{3}{9}}$ as $\sqrt[9]{{(3 m)}^{3}}$ .

$\frac{9 m (\sqrt[9]{{(3 m)}^{3}} \sqrt[9]{729 m^{7}})}{27 m}$

Step 4.7.2

Combine using the product rule for radicals.

$\frac{9 m \sqrt[9]{{(3 m)}^{3} (729 m^{7})}}{27 m}$

Step 4.7.3

Apply the product rule to $3 m$ .

$\frac{9 m \sqrt[9]{3^{3} m^{3} \cdot 729 m^{7}}}{27 m}$

Step 4.7.4

Combine exponents.

Tap for more steps...

Step 4.7.4.1

Rewrite $729$ as $3^{6}$ .

$\frac{9 m \sqrt[9]{3^{6} \cdot 3^{3} m^{3} m^{7}}}{27 m}$

Step 4.7.4.2

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

$\frac{9 m \sqrt[9]{3^{6 + 3} m^{3} m^{7}}}{27 m}$

Step 4.7.4.3

Add $6$ and $3$ .

$\frac{9 m \sqrt[9]{3^{9} m^{3} m^{7}}}{27 m}$

Step 4.7.4.4

Multiply $m^{3}$ by $m^{7}$ by adding the exponents.

Tap for more steps...

Step 4.7.4.4.1

Move $m^{7}$ .

$\frac{9 m \sqrt[9]{3^{9} (m^{7} m^{3})}}{27 m}$

Step 4.7.4.4.2

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

$\frac{9 m \sqrt[9]{3^{9} m^{7 + 3}}}{27 m}$

Step 4.7.4.4.3

Add $7$ and $3$ .

$\frac{9 m \sqrt[9]{3^{9} m^{10}}}{27 m}$

Step 4.7.5

Raise $3$ to the power of $9$ .

$\frac{9 m \sqrt[9]{19683 m^{10}}}{27 m}$

Step 4.8

Rewrite $19683 m^{10}$ as ${(3 m)}^{9} m$ .

Tap for more steps...

Step 4.8.1

Rewrite $19683$ as $3^{9}$ .

$\frac{9 m \sqrt[9]{3^{9} m^{10}}}{27 m}$

Step 4.8.2

Factor out $m^{9}$ .

$\frac{9 m \sqrt[9]{3^{9} (m^{9} m)}}{27 m}$

Step 4.8.3

Rewrite $3^{9} m^{9}$ as ${(3 m)}^{9}$ .

$\frac{9 m \sqrt[9]{{(3 m)}^{9} m}}{27 m}$

Step 4.9

Pull terms out from under the radical.

$\frac{9 m (3 m \sqrt[9]{m})}{27 m}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

Multiply $3$ by $9$ .

$\frac{27 m (m \sqrt[9]{m})}{27 m}$

Step 5.2

Raise $m$ to the power of $1$ .

$\frac{27 (m^{1} m) \sqrt[9]{m}}{27 m}$

Step 5.3

Raise $m$ to the power of $1$ .

$\frac{27 (m^{1} m^{1}) \sqrt[9]{m}}{27 m}$

Step 5.4

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

$\frac{27 m^{1 + 1} \sqrt[9]{m}}{27 m}$

Step 5.5

Add $1$ and $1$ .

$\frac{27 m^{2} \sqrt[9]{m}}{27 m}$

Step 6

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.1

Cancel the common factor of $27$ .

Tap for more steps...

Step 6.1.1

Cancel the common factor.

$\frac{27 m^{2} \sqrt[9]{m}}{27 m}$

Step 6.1.2

Rewrite the expression.

$\frac{m^{2} \sqrt[9]{m}}{m}$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Factor $m$ out of $m^{2} \sqrt[9]{m}$ .

$\frac{m (m \sqrt[9]{m})}{m}$

Step 6.2.2

Cancel the common factors.

Tap for more steps...

Step 6.2.2.1

Raise $m$ to the power of $1$ .

$\frac{m (m \sqrt[9]{m})}{m^{1}}$

Step 6.2.2.2

Factor $m$ out of $m^{1}$ .

$\frac{m (m \sqrt[9]{m})}{m \cdot 1}$

Step 6.2.2.3

Cancel the common factor.

$\frac{m (m \sqrt[9]{m})}{m \cdot 1}$

Step 6.2.2.4

Rewrite the expression.

$\frac{m \sqrt[9]{m}}{1}$

Step 6.2.2.5

Divide $m \sqrt[9]{m}$ by $1$ .

$m \sqrt[9]{m}$