Basic Math Examples

$- 9 {(2 m^{6} n^{4})}^{2} - {(- m^{5} n^{3})}^{2}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to $2 m^{6} n^{4}$ .

$- 9 ({(2 m^{6})}^{2} {(n^{4})}^{2}) - {(- m^{5} n^{3})}^{2}$

Step 1.2

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

$- 9 (2^{2} {(m^{6})}^{2} {(n^{4})}^{2}) - {(- m^{5} n^{3})}^{2}$

Step 2

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

$- 9 (4 {(m^{6})}^{2} {(n^{4})}^{2}) - {(- m^{5} n^{3})}^{2}$

Step 3

Multiply the exponents in ${(m^{6})}^{2}$ .

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Step 3.1

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

$- 9 (4 m^{6 \cdot 2} {(n^{4})}^{2}) - {(- m^{5} n^{3})}^{2}$

Step 3.2

Multiply $6$ by $2$ .

$- 9 (4 m^{12} {(n^{4})}^{2}) - {(- m^{5} n^{3})}^{2}$

Step 4

Multiply the exponents in ${(n^{4})}^{2}$ .

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Step 4.1

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

$- 9 (4 m^{12} n^{4 \cdot 2}) - {(- m^{5} n^{3})}^{2}$

Step 4.2

Multiply $4$ by $2$ .

$- 9 (4 m^{12} n^{8}) - {(- m^{5} n^{3})}^{2}$

Step 5

Multiply $4$ by $- 9$ .

$- 36 (m^{12} n^{8}) - {(- m^{5} n^{3})}^{2}$

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to $- m^{5} n^{3}$ .

$- 36 m^{12} n^{8} - ({(- m^{5})}^{2} {(n^{3})}^{2})$

Step 6.2

Apply the product rule to $- m^{5}$ .

$- 36 m^{12} n^{8} - ({(- 1)}^{2} {(m^{5})}^{2} {(n^{3})}^{2})$

Step 7

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 7.1

Move ${(- 1)}^{2}$ .

$- 36 m^{12} n^{8} + {(- 1)}^{2} \cdot - 1 ({(m^{5})}^{2} {(n^{3})}^{2})$

Step 7.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 7.2.1

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

$- 36 m^{12} n^{8} + {(- 1)}^{2} \cdot {(- 1)}^{1} ({(m^{5})}^{2} {(n^{3})}^{2})$

Step 7.2.2

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

$- 36 m^{12} n^{8} + {(- 1)}^{2 + 1} ({(m^{5})}^{2} {(n^{3})}^{2})$

Step 7.3

Add $2$ and $1$ .

$- 36 m^{12} n^{8} + {(- 1)}^{3} ({(m^{5})}^{2} {(n^{3})}^{2})$

Step 8

Raise $- 1$ to the power of $3$ .

$- 36 m^{12} n^{8} - ({(m^{5})}^{2} {(n^{3})}^{2})$

Step 9

Multiply the exponents in ${(m^{5})}^{2}$ .

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Step 9.1

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

$- 36 m^{12} n^{8} - (m^{5 \cdot 2} {(n^{3})}^{2})$

Step 9.2

Multiply $5$ by $2$ .

$- 36 m^{12} n^{8} - (m^{10} {(n^{3})}^{2})$

Step 10

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

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Step 10.1

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

$- 36 m^{12} n^{8} - (m^{10} n^{3 \cdot 2})$

Step 10.2

Multiply $3$ by $2$ .

$- 36 m^{12} n^{8} - (m^{10} n^{6})$

$- 36 m^{12} n^{8} - m^{10} n^{6}$