Finite Math Examples

$\frac{3 m^{8} n^{8} + 15 m^{7} n^{6} + 21 m^{11} n^{7}}{3 m^{2} n}$

Step 1

Factor $3 m^{7} n^{6}$ out of $3 m^{8} n^{8} + 15 m^{7} n^{6} + 21 m^{11} n^{7}$ .

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Step 1.1

Factor $3 m^{7} n^{6}$ out of $3 m^{8} n^{8}$ .

$\frac{3 m^{7} n^{6} (m n^{2}) + 15 m^{7} n^{6} + 21 m^{11} n^{7}}{3 m^{2} n}$

Step 1.2

Factor $3 m^{7} n^{6}$ out of $15 m^{7} n^{6}$ .

$\frac{3 m^{7} n^{6} (m n^{2}) + 3 m^{7} n^{6} (5) + 21 m^{11} n^{7}}{3 m^{2} n}$

Step 1.3

Factor $3 m^{7} n^{6}$ out of $21 m^{11} n^{7}$ .

$\frac{3 m^{7} n^{6} (m n^{2}) + 3 m^{7} n^{6} (5) + 3 m^{7} n^{6} (7 m^{4} n)}{3 m^{2} n}$

Step 1.4

Factor $3 m^{7} n^{6}$ out of $3 m^{7} n^{6} (m n^{2}) + 3 m^{7} n^{6} (5)$ .

$\frac{3 m^{7} n^{6} (m n^{2} + 5) + 3 m^{7} n^{6} (7 m^{4} n)}{3 m^{2} n}$

Step 1.5

Factor $3 m^{7} n^{6}$ out of $3 m^{7} n^{6} (m n^{2} + 5) + 3 m^{7} n^{6} (7 m^{4} n)$ .

$\frac{3 m^{7} n^{6} (m n^{2} + 5 + 7 m^{4} n)}{3 m^{2} n}$

Step 2

Cancel the common factor of $3$ .

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Step 2.1

Cancel the common factor.

$\frac{3 m^{7} n^{6} (m n^{2} + 5 + 7 m^{4} n)}{3 m^{2} n}$

Step 2.2

Rewrite the expression.

$\frac{m^{7} n^{6} (m n^{2} + 5 + 7 m^{4} n)}{m^{2} n}$

Step 3

Cancel the common factor of $m^{7}$ and $m^{2}$ .

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Step 3.1

Factor $m^{2}$ out of $m^{7} n^{6} (m n^{2} + 5 + 7 m^{4} n)$ .

$\frac{m^{2} (m^{5} n^{6} (m n^{2} + 5 + 7 m^{4} n))}{m^{2} n}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\frac{m^{2} (m^{5} n^{6} (m n^{2} + 5 + 7 m^{4} n))}{m^{2} (n)}$

Step 3.2.2

Cancel the common factor.

$\frac{m^{2} (m^{5} n^{6} (m n^{2} + 5 + 7 m^{4} n))}{m^{2} n}$

Step 3.2.3

Rewrite the expression.

$\frac{m^{5} n^{6} (m n^{2} + 5 + 7 m^{4} n)}{n}$

Step 4

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

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Step 4.1

Factor $n$ out of $m^{5} n^{6} (m n^{2} + 5 + 7 m^{4} n)$ .

$\frac{n (m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n))}{n}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{n (m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n))}{n^{1}}$

Step 4.2.2

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

$\frac{n (m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n))}{n \cdot 1}$

Step 4.2.3

Cancel the common factor.

$\frac{n (m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n))}{n \cdot 1}$

Step 4.2.4

Rewrite the expression.

$\frac{m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n)}{1}$

Step 4.2.5

Divide $m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n)$ by $1$ .

$m^{5} n^{5} (m n^{2} + 5 + 7 m^{4} n)$

Step 5

Apply the distributive property.

$m^{5} n^{5} (m n^{2}) + m^{5} n^{5} \cdot 5 + m^{5} n^{5} (7 m^{4} n)$

Step 6

Simplify.

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Step 6.1

Multiply $m^{5}$ by $m$ by adding the exponents.

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Step 6.1.1

Move $m$ .

$m \cdot m^{5} n^{5} n^{2} + m^{5} n^{5} \cdot 5 + m^{5} n^{5} (7 m^{4} n)$

Step 6.1.2

Multiply $m$ by $m^{5}$ .

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Step 6.1.2.1

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

$m^{1} m^{5} n^{5} n^{2} + m^{5} n^{5} \cdot 5 + m^{5} n^{5} (7 m^{4} n)$

Step 6.1.2.2

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

$m^{1 + 5} n^{5} n^{2} + m^{5} n^{5} \cdot 5 + m^{5} n^{5} (7 m^{4} n)$

Step 6.1.3

Add $1$ and $5$ .

$m^{6} n^{5} n^{2} + m^{5} n^{5} \cdot 5 + m^{5} n^{5} (7 m^{4} n)$

Step 6.2

Move $5$ to the left of $m^{5} n^{5}$ .

$m^{6} n^{5} n^{2} + 5 \cdot (m^{5} n^{5}) + m^{5} n^{5} (7 m^{4} n)$

Step 6.3

Rewrite using the commutative property of multiplication.

$m^{6} n^{5} n^{2} + 5 \cdot (m^{5} n^{5}) + 7 m^{5} n^{5} (m^{4} n)$

Step 7

Simplify each term.

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Step 7.1

Multiply $n^{5}$ by $n^{2}$ by adding the exponents.

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Step 7.1.1

Move $n^{2}$ .

$m^{6} (n^{2} n^{5}) + 5 \cdot (m^{5} n^{5}) + 7 m^{5} n^{5} (m^{4} n)$

Step 7.1.2

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

$m^{6} n^{2 + 5} + 5 \cdot (m^{5} n^{5}) + 7 m^{5} n^{5} (m^{4} n)$

Step 7.1.3

Add $2$ and $5$ .

$m^{6} n^{7} + 5 \cdot (m^{5} n^{5}) + 7 m^{5} n^{5} (m^{4} n)$

Step 7.2

Multiply $m^{5}$ by $m^{4}$ by adding the exponents.

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Step 7.2.1

Move $m^{4}$ .

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 (m^{4} m^{5}) n^{5} n$

Step 7.2.2

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

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{4 + 5} n^{5} n$

Step 7.2.3

Add $4$ and $5$ .

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{9} n^{5} n$

Step 7.3

Multiply $n^{5}$ by $n$ by adding the exponents.

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Step 7.3.1

Move $n$ .

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{9} (n \cdot n^{5})$

Step 7.3.2

Multiply $n$ by $n^{5}$ .

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Step 7.3.2.1

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

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{9} (n^{1} n^{5})$

Step 7.3.2.2

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

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{9} n^{1 + 5}$

Step 7.3.3

Add $1$ and $5$ .

$m^{6} n^{7} + 5 m^{5} n^{5} + 7 m^{9} n^{6}$