Algebra Examples

$\frac{2 m n^{2} {(3 m^{2} n)}^{2}}{12 m^{3} {(n^{4})}^{- 3}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Move ${(n^{4})}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{2 m n^{2} {(3 m^{2} n)}^{2} {(n^{4})}^{3}}{12 m^{3}}$

Step 1.2

Cancel the common factor of $2$ and $12$ .

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Step 1.2.1

Factor $2$ out of $2 m n^{2} {(3 m^{2} n)}^{2} {(n^{4})}^{3}$ .

$\frac{2 (m n^{2} {(3 m^{2} n)}^{2} {(n^{4})}^{3})}{12 m^{3}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $12 m^{3}$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.3

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

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Step 1.3.1

Factor $m$ out of $m n^{2} {(3 m^{2} n)}^{2} {(n^{4})}^{3}$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $m$ out of $6 m^{3}$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Move $n^{2}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Add $2$ and $2$ .

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

Step 2.3

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

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

Step 2.4

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

$\frac{n^{4} (9 {(m^{2})}^{2}) {(n^{4})}^{3}}{6 m^{2}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{n^{4} (9 m^{2 \cdot 2}) {(n^{4})}^{3}}{6 m^{2}}$

Step 2.5.2

Multiply $2$ by $2$ .

$\frac{n^{4} (9 m^{4}) {(n^{4})}^{3}}{6 m^{2}}$

Step 2.6

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

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Step 2.6.1

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

$\frac{n^{4} (9 m^{4}) n^{4 \cdot 3}}{6 m^{2}}$

Step 2.6.2

Multiply $4$ by $3$ .

$\frac{n^{4} (9 m^{4}) n^{12}}{6 m^{2}}$

$\frac{n^{4} \cdot 9 m^{4} n^{12}}{6 m^{2}}$

Step 2.7

Multiply $n^{4}$ by $n^{12}$ by adding the exponents.

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Step 2.7.1

Move $n^{12}$ .

$\frac{n^{12} n^{4} \cdot 9 m^{4}}{6 m^{2}}$

Step 2.7.2

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

$\frac{n^{12 + 4} \cdot 9 m^{4}}{6 m^{2}}$

Step 2.7.3

Add $12$ and $4$ .

$\frac{n^{16} \cdot 9 m^{4}}{6 m^{2}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $9$ and $6$ .

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Step 3.1.1

Factor $3$ out of $n^{16} \cdot 9 m^{4}$ .

$\frac{3 (n^{16} \cdot 3 m^{4})}{6 m^{2}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $3$ out of $6 m^{2}$ .

$\frac{3 (n^{16} \cdot 3 m^{4})}{3 (2 m^{2})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{3 (n^{16} \cdot 3 m^{4})}{3 (2 m^{2})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{n^{16} \cdot 3 m^{4}}{2 m^{2}}$

Step 3.2

Cancel the common factor of $m^{4}$ and $m^{2}$ .

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Step 3.2.1

Factor $m^{2}$ out of $n^{16} \cdot 3 m^{4}$ .

$\frac{m^{2} (n^{16} \cdot 3 m^{2})}{2 m^{2}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

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

$\frac{m^{2} (n^{16} \cdot 3 m^{2})}{m^{2} \cdot 2}$

Step 3.2.2.2

Cancel the common factor.

$\frac{m^{2} (n^{16} \cdot 3 m^{2})}{m^{2} \cdot 2}$

Step 3.2.2.3

Rewrite the expression.

$\frac{n^{16} \cdot 3 m^{2}}{2}$

Step 3.3

Move $3$ to the left of $n^{16}$ .

$\frac{3 n^{16} m^{2}}{2}$