Algebra Examples

$\frac{- 4 {(m^{- 3} z^{4})}^{2}}{12 m^{- 4} z^{- 1}}$

Step 1

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

$\frac{- 4 {(m^{- 3} z^{4})}^{2} m^{4}}{12 z^{- 1}}$

Step 2

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

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

Step 3

Cancel the common factor of $- 4$ and $12$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $4$ out of $12$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the product rule to $m^{- 3} z^{4}$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $- 3$ by $2$ .

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

$\frac{- \frac{1}{m^{6}} z^{4 \cdot 2} m^{4} z}{3}$

Step 4.4.2

Multiply $4$ by $2$ .

$\frac{- \frac{1}{m^{6}} z^{8} m^{4} z}{3}$

Step 4.5

Combine exponents.

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Step 4.5.1

Combine $z^{8}$ and $\frac{1}{m^{6}}$ .

$\frac{- \frac{z^{8}}{m^{6}} m^{4} z}{3}$

Step 4.5.2

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

$\frac{- \frac{m^{4} z^{8}}{m^{6}} z}{3}$

Step 4.5.3

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

$\frac{- \frac{z (m^{4} z^{8})}{m^{6}}}{3}$

Step 4.5.4

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

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

Step 4.5.5

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

$\frac{- \frac{m^{4} z^{1 + 8}}{m^{6}}}{3}$

Step 4.5.6

Add $1$ and $8$ .

$\frac{- \frac{m^{4} z^{9}}{m^{6}}}{3}$

Step 4.6

Reduce the expression $\frac{m^{4} z^{9}}{m^{6}}$ by cancelling the common factors.

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Step 4.6.1

Factor $m^{4}$ out of $m^{4} z^{9}$ .

$\frac{- \frac{m^{4} (z^{9})}{m^{6}}}{3}$

Step 4.6.2

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

$\frac{- \frac{m^{4} (z^{9})}{m^{4} m^{2}}}{3}$

Step 4.6.3

Cancel the common factor.

$\frac{- \frac{m^{4} z^{9}}{m^{4} m^{2}}}{3}$

Step 4.6.4

Rewrite the expression.

$\frac{- \frac{z^{9}}{m^{2}}}{3}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$- \frac{z^{9}}{m^{2}} \cdot \frac{1}{3}$

Step 6

Multiply $\frac{1}{3}$ by $\frac{z^{9}}{m^{2}}$ .

$- \frac{z^{9}}{3 m^{2}}$