Algebra Examples

${(- 4 m^{2} n)}^{4} \cdot \frac{1}{6} m^{- 10} n^{- 4}$

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 $- 4 m^{2} n$ .

${(- 4 m^{2})}^{4} n^{4} \cdot \frac{1}{6} \cdot (m^{- 10} n^{- 4})$

Step 1.2

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

${(- 4)}^{4} {(m^{2})}^{4} n^{4} \cdot \frac{1}{6} \cdot (m^{- 10} n^{- 4})$

Step 2

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

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Step 2.1

Move $n^{- 4}$ .

${(- 4)}^{4} {(m^{2})}^{4} (n^{- 4} n^{4}) \cdot \frac{1}{6} \cdot m^{- 10}$

Step 2.2

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

${(- 4)}^{4} {(m^{2})}^{4} n^{- 4 + 4} \cdot \frac{1}{6} \cdot m^{- 10}$

Step 2.3

Add $- 4$ and $4$ .

${(- 4)}^{4} {(m^{2})}^{4} n^{0} \cdot \frac{1}{6} \cdot m^{- 10}$

Step 3

Simplify the expression.

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Step 3.1

Simplify ${(- 4)}^{4} {(m^{2})}^{4} n^{0} \cdot \frac{1}{6} \cdot m^{- 10}$ .

${(- 4)}^{4} {(m^{2})}^{4} \cdot \frac{1}{6} \cdot m^{- 10}$

Step 3.2

Raise $- 4$ to the power of $4$ .

$256 {(m^{2})}^{4} \cdot \frac{1}{6} \cdot m^{- 10}$

Step 4

Cancel the common factor of $2$ .

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Step 4.1

Factor $2$ out of $256 {(m^{2})}^{4}$ .

$2 (128 {(m^{2})}^{4}) \cdot \frac{1}{6} \cdot m^{- 10}$

Step 4.2

Factor $2$ out of $6$ .

$2 (128 {(m^{2})}^{4}) \cdot \frac{1}{2 (3)} \cdot m^{- 10}$

Step 4.3

Cancel the common factor.

$2 (128 {(m^{2})}^{4}) \cdot \frac{1}{2 \cdot 3} \cdot m^{- 10}$

Step 4.4

Rewrite the expression.

$128 {(m^{2})}^{4} \cdot \frac{1}{3} \cdot m^{- 10}$

Step 5

Combine $128$ and $\frac{1}{3}$ .

${(m^{2})}^{4} \cdot \frac{128}{3} \cdot m^{- 10}$

Step 6

Combine ${(m^{2})}^{4}$ and $\frac{128}{3}$ .

$\frac{{(m^{2})}^{4} \cdot 128}{3} \cdot m^{- 10}$

Step 7

Simplify the expression.

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Step 7.1

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

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Step 7.1.1

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

$\frac{m^{2 \cdot 4} \cdot 128}{3} \cdot m^{- 10}$

Step 7.1.2

Multiply $2$ by $4$ .

$\frac{m^{8} \cdot 128}{3} \cdot m^{- 10}$

Step 7.2

Move $128$ to the left of $m^{8}$ .

$\frac{128 \cdot m^{8}}{3} \cdot m^{- 10}$

Step 8

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

$\frac{128 m^{8}}{3} \cdot \frac{1}{m^{10}}$

Step 9

Combine.

$\frac{128 m^{8} \cdot 1}{3 m^{10}}$

Step 10

Cancel the common factor of $m^{8}$ and $m^{10}$ .

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Step 10.1

Factor $m^{8}$ out of $128 m^{8} \cdot 1$ .

$\frac{m^{8} (128 \cdot 1)}{3 m^{10}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $m^{8}$ out of $3 m^{10}$ .

$\frac{m^{8} (128 \cdot 1)}{m^{8} (3 m^{2})}$

Step 10.2.2

Cancel the common factor.

$\frac{m^{8} (128 \cdot 1)}{m^{8} (3 m^{2})}$

Step 10.2.3

Rewrite the expression.

$\frac{128 \cdot 1}{3 m^{2}}$

Step 11

Multiply $128$ by $1$ .

$\frac{128}{3 m^{2}}$