Basic Math Examples

$\frac{{(2 m^{2} n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(2 m^{2})}^{2} {(n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.2

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

$\frac{2^{2} {(m^{2})}^{2} {(n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.3

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

$\frac{4 {(m^{2})}^{2} {(n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{4 m^{2 \cdot 2} {(n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.4.2

Multiply $2$ by $2$ .

$\frac{4 m^{4} {(n^{3})}^{2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.5

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

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Step 1.5.1

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

$\frac{4 m^{4} n^{3 \cdot 2}}{{(m^{2} n^{- 2})}^{2} \cdot {(2 m^{2} n^{4})}^{3}}$

Step 1.5.2

Multiply $3$ by $2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{4 m^{4} n^{6}}{{(m^{2})}^{2} {(n^{- 2})}^{2} {(2 m^{2} n^{4})}^{3}}$

Step 2.2

Apply the product rule to $2 m^{2} n^{4}$ .

$\frac{4 m^{4} n^{6}}{{(m^{2})}^{2} {(n^{- 2})}^{2} {(2 m^{2})}^{3} {(n^{4})}^{3}}$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $2$ by $2$ .

$\frac{4 m^{4} n^{6}}{m^{4} {(n^{- 2})}^{2} {(2 m^{2})}^{3} {(n^{4})}^{3}}$

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Multiply $- 2$ by $2$ .

$\frac{4 m^{4} n^{6}}{m^{4} n^{- 4} {(2 m^{2})}^{3} {(n^{4})}^{3}}$

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Raise $2$ to the power of $3$ .

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} (8 {(m^{2})}^{3}) {(n^{4})}^{3}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} (8 m^{2 \cdot 3}) {(n^{4})}^{3}}$

Step 2.8.2

Multiply $2$ by $3$ .

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} (8 m^{6}) {(n^{4})}^{3}}$

Step 2.9

Multiply the exponents in ${(n^{4})}^{3}$ .

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Step 2.9.1

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

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} (8 m^{6}) n^{4 \cdot 3}}$

Step 2.9.2

Multiply $4$ by $3$ .

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} (8 m^{6}) n^{12}}$

$\frac{4 m^{4} n^{6}}{m^{4} \frac{1}{n^{4}} \cdot 8 m^{6} n^{12}}$

Step 2.10

Combine exponents.

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Step 2.10.1

Combine $m^{4}$ and $\frac{1}{n^{4}}$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{4}}{n^{4}} \cdot 8 m^{6} n^{12}}$

Step 2.10.2

Combine $\frac{m^{4}}{n^{4}}$ and $8$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{4} \cdot 8}{n^{4}} m^{6} n^{12}}$

Step 2.10.3

Combine $\frac{m^{4} \cdot 8}{n^{4}}$ and $m^{6}$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{4} \cdot 8 m^{6}}{n^{4}} n^{12}}$

Step 2.10.4

Multiply $m^{4}$ by $m^{6}$ by adding the exponents.

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Step 2.10.4.1

Move $m^{6}$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{6} m^{4} \cdot 8}{n^{4}} n^{12}}$

Step 2.10.4.2

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

$\frac{4 m^{4} n^{6}}{\frac{m^{6 + 4} \cdot 8}{n^{4}} n^{12}}$

Step 2.10.4.3

Add $6$ and $4$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{10} \cdot 8}{n^{4}} n^{12}}$

Step 2.10.5

Combine $\frac{m^{10} \cdot 8}{n^{4}}$ and $n^{12}$ .

$\frac{4 m^{4} n^{6}}{\frac{m^{10} \cdot 8 n^{12}}{n^{4}}}$

Step 2.11

Reduce the expression by cancelling the common factors.

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Step 2.11.1

Reduce the expression $\frac{m^{10} \cdot 8 n^{12}}{n^{4}}$ by cancelling the common factors.

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Step 2.11.1.1

Factor $n^{4}$ out of $m^{10} \cdot 8 n^{12}$ .

$\frac{4 m^{4} n^{6}}{\frac{n^{4} (m^{10} \cdot 8 n^{8})}{n^{4}}}$

Step 2.11.1.2

Multiply by $1$ .

$\frac{4 m^{4} n^{6}}{\frac{n^{4} (m^{10} \cdot 8 n^{8})}{n^{4} \cdot 1}}$

Step 2.11.1.3

Cancel the common factor.

$\frac{4 m^{4} n^{6}}{\frac{n^{4} (m^{10} \cdot 8 n^{8})}{n^{4} \cdot 1}}$

Step 2.11.1.4

Rewrite the expression.

$\frac{4 m^{4} n^{6}}{\frac{m^{10} \cdot 8 n^{8}}{1}}$

Step 2.11.2

Divide $m^{10} \cdot 8 n^{8}$ by $1$ .

$\frac{4 m^{4} n^{6}}{m^{10} \cdot 8 n^{8}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $4$ and $8$ .

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Step 3.1.1

Factor $4$ out of $4 m^{4} n^{6}$ .

$\frac{4 (m^{4} n^{6})}{m^{10} \cdot 8 n^{8}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $4$ out of $m^{10} \cdot 8 n^{8}$ .

$\frac{4 (m^{4} n^{6})}{4 (m^{10} \cdot 2 n^{8})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{4 (m^{4} n^{6})}{4 (m^{10} \cdot 2 n^{8})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{m^{4} n^{6}}{m^{10} \cdot 2 n^{8}}$

Step 3.2

Cancel the common factor of $m^{4}$ and $m^{10}$ .

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Step 3.2.1

Factor $m^{4}$ out of $m^{4} n^{6}$ .

$\frac{m^{4} (n^{6})}{m^{10} \cdot 2 n^{8}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $m^{4}$ out of $m^{10} \cdot 2 n^{8}$ .

$\frac{m^{4} (n^{6})}{m^{4} (m^{6} \cdot 2 n^{8})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{m^{4} n^{6}}{m^{4} (m^{6} \cdot 2 n^{8})}$

Step 3.2.2.3

Rewrite the expression.

$\frac{n^{6}}{m^{6} \cdot 2 n^{8}}$

Step 3.3

Cancel the common factor of $n^{6}$ and $n^{8}$ .

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Step 3.3.1

Multiply by $1$ .

$\frac{n^{6} \cdot 1}{m^{6} \cdot 2 n^{8}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $n^{6}$ out of $m^{6} \cdot 2 n^{8}$ .

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

Move $2$ to the left of $m^{6}$ .

$\frac{1}{2 m^{6} n^{2}}$