Basic Math Examples

$\frac{{(2 m^{- 2} n^{3})}^{- 2}}{{(6 m^{- 1} n^{- 2})}^{- 3}}$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Move ${(2 m^{- 2} n^{3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(6 m^{- 1} n^{- 2})}^{- 3} {(2 m^{- 2} n^{3})}^{2}}$

Step 1.2

Move ${(6 m^{- 1} n^{- 2})}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(6 m^{- 1} n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2

Simplify the numerator.

Tap for more steps...

Step 2.1

Apply the product rule to $6 m^{- 1} n^{- 2}$ .

$\frac{{(6 m^{- 1})}^{3} {(n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(6 \frac{1}{m})}^{3} {(n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.3

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

$\frac{{(\frac{6}{m})}^{3} {(n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.4

Apply the product rule to $\frac{6}{m}$ .

$\frac{\frac{6^{3}}{m^{3}} {(n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.5

Raise $6$ to the power of $3$ .

$\frac{\frac{216}{m^{3}} {(n^{- 2})}^{3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.6

Multiply the exponents in ${(n^{- 2})}^{3}$ .

Tap for more steps...

Step 2.6.1

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

$\frac{\frac{216}{m^{3}} n^{- 2 \cdot 3}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.6.2

Multiply $- 2$ by $3$ .

$\frac{\frac{216}{m^{3}} n^{- 6}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 2.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{{(2 m^{- 2} n^{3})}^{2}}$

Step 3

Simplify the denominator.

Tap for more steps...

Step 3.1

Apply the product rule to $2 m^{- 2} n^{3}$ .

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{{(2 m^{- 2})}^{2} {(n^{3})}^{2}}$

Step 3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{{(2 \frac{1}{m^{2}})}^{2} {(n^{3})}^{2}}$

Step 3.3

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

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{{(\frac{2}{m^{2}})}^{2} {(n^{3})}^{2}}$

Step 3.4

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

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{2^{2}}{{(m^{2})}^{2}} {(n^{3})}^{2}}$

Step 3.5

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

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{4}{{(m^{2})}^{2}} {(n^{3})}^{2}}$

Step 3.6

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

Tap for more steps...

Step 3.6.1

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

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{4}{m^{2 \cdot 2}} {(n^{3})}^{2}}$

Step 3.6.2

Multiply $2$ by $2$ .

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{4}{m^{4}} {(n^{3})}^{2}}$

Step 3.7

Multiply the exponents in ${(n^{3})}^{2}$ .

Tap for more steps...

Step 3.7.1

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

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{4}{m^{4}} n^{3 \cdot 2}}$

Step 3.7.2

Multiply $3$ by $2$ .

$\frac{\frac{216}{m^{3}} \cdot \frac{1}{n^{6}}}{\frac{4}{m^{4}} n^{6}}$

Step 4

Combine fractions.

Tap for more steps...

Step 4.1

Multiply $\frac{216}{m^{3}}$ by $\frac{1}{n^{6}}$ .

$\frac{\frac{216}{m^{3} n^{6}}}{\frac{4}{m^{4}} n^{6}}$

Step 4.2

Combine $\frac{4}{m^{4}}$ and $n^{6}$ .

$\frac{\frac{216}{m^{3} n^{6}}}{\frac{4 n^{6}}{m^{4}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{216}{m^{3} n^{6}} \cdot \frac{m^{4}}{4 n^{6}}$

Step 6

Combine.

$\frac{216 m^{4}}{m^{3} n^{6} (4 n^{6})}$

Step 7

Multiply $n^{6}$ by $n^{6}$ by adding the exponents.

Tap for more steps...

Step 7.1

Move $n^{6}$ .

$\frac{216 m^{4}}{m^{3} (n^{6} n^{6}) \cdot 4}$

Step 7.2

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

$\frac{216 m^{4}}{m^{3} n^{6 + 6} \cdot 4}$

Step 7.3

Add $6$ and $6$ .

$\frac{216 m^{4}}{m^{3} n^{12} \cdot 4}$

Step 8

Cancel the common factor of $216$ and $4$ .

Tap for more steps...

Step 8.1

Factor $4$ out of $216 m^{4}$ .

$\frac{4 (54 m^{4})}{m^{3} n^{12} \cdot 4}$

Step 8.2

Cancel the common factors.

Tap for more steps...

Step 8.2.1

Factor $4$ out of $m^{3} n^{12} \cdot 4$ .

$\frac{4 (54 m^{4})}{4 \cdot (m^{3} n^{12})}$

Step 8.2.2

Cancel the common factor.

$\frac{4 (54 m^{4})}{4 \cdot (m^{3} n^{12})}$

Step 8.2.3

Rewrite the expression.

$\frac{54 m^{4}}{m^{3} n^{12}}$

Step 9

Cancel the common factor of $m^{4}$ and $m^{3}$ .

Tap for more steps...

Step 9.1

Factor $m^{3}$ out of $54 m^{4}$ .

$\frac{m^{3} (54 m)}{m^{3} n^{12}}$

Step 9.2

Cancel the common factors.

Tap for more steps...

Step 9.2.1

Factor $m^{3}$ out of $m^{3} n^{12}$ .

$\frac{m^{3} (54 m)}{m^{3} (n^{12})}$

Step 9.2.2

Cancel the common factor.

$\frac{m^{3} (54 m)}{m^{3} n^{12}}$

Step 9.2.3

Rewrite the expression.

$\frac{54 m}{n^{12}}$