Simplify (16/(m-3)-4/(m-4))/(16/(m^2)-(m-4)/(m-3))

$\frac{\frac{16}{m - 3} - \frac{4}{m - 4}}{\frac{16}{m^{2}} - \frac{m - 4}{m - 3}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $(m - 3) (m - 4) m^{2}$ .

Multiply $\frac{\frac{16}{m - 3} - \frac{4}{m - 4}}{\frac{16}{m^{2}} - \frac{m - 4}{m - 3}}$ by $\frac{(m - 3) (m - 4) m^{2}}{(m - 3) (m - 4) m^{2}}$ .

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

Combine.

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

Step 2

Apply the distributive property.

$\frac{(m - 3) (m - 4) m^{2} \frac{16}{m - 3} + (m - 3) (m - 4) m^{2} (- \frac{4}{m - 4})}{(m - 3) (m - 4) m^{2} \frac{16}{m^{2}} + (m - 3) (m - 4) m^{2} (- \frac{m - 4}{m - 3})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $m - 3$ .

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

$\frac{(m - 3) ((m - 4) m^{2}) \frac{16}{m - 3} + (m - 3) (m - 4) m^{2} (- \frac{4}{m - 4})}{(m - 3) (m - 4) m^{2} \frac{16}{m^{2}} + (m - 3) (m - 4) m^{2} (- \frac{m - 4}{m - 3})}$

Cancel the common factor.

$\frac{(m - 3) ((m - 4) m^{2}) \frac{16}{m - 3} + (m - 3) (m - 4) m^{2} (- \frac{4}{m - 4})}{(m - 3) (m - 4) m^{2} \frac{16}{m^{2}} + (m - 3) (m - 4) m^{2} (- \frac{m - 4}{m - 3})}$

Rewrite the expression.

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

Cancel the common factor of $m - 4$ .

Move the leading negative in $- \frac{4}{m - 4}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $m - 3$ .

Move the leading negative in $- \frac{m - 4}{m - 3}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Raise $m - 4$ to the power of $1$ .

$\frac{(m - 4) m^{2} \cdot 16 + (m - 3) m^{2} \cdot - 4}{(m - 3) (m - 4) \cdot 16 + m^{2} (- ({(m - 4)}^{1} (m - 4)))}$

Raise $m - 4$ to the power of $1$ .

$\frac{(m - 4) m^{2} \cdot 16 + (m - 3) m^{2} \cdot - 4}{(m - 3) (m - 4) \cdot 16 + m^{2} (- ({(m - 4)}^{1} {(m - 4)}^{1}))}$

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

$\frac{(m - 4) m^{2} \cdot 16 + (m - 3) m^{2} \cdot - 4}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{1 + 1})}$

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

Factor $m^{2} \cdot 4$ out of $(m - 4) m^{2} \cdot 16 + (m - 3) m^{2} \cdot - 4$ .

Factor $m^{2} \cdot 4$ out of $(m - 4) m^{2} \cdot 16$ .

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

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

$\frac{m^{2} \cdot 4 ((m - 4) \cdot 4) + m^{2} \cdot 4 ((m - 3) \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Factor $m^{2} \cdot 4$ out of $m^{2} \cdot 4 ((m - 4) \cdot 4) + m^{2} \cdot 4 ((m - 3) \cdot - 1)$ .

$\frac{m^{2} \cdot 4 ((m - 4) \cdot 4 + (m - 3) \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Apply the distributive property.

$\frac{m^{2} \cdot 4 (m \cdot 4 - 4 \cdot 4 + (m - 3) \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Move $4$ to the left of $m$ .

$\frac{m^{2} \cdot 4 (4 \cdot m - 4 \cdot 4 + (m - 3) \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Multiply $- 4$ by $4$ .

$\frac{m^{2} \cdot 4 (4 \cdot m - 16 + (m - 3) \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Apply the distributive property.

$\frac{m^{2} \cdot 4 (4 m - 16 + m \cdot - 1 - 3 \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Move $- 1$ to the left of $m$ .

$\frac{m^{2} \cdot 4 (4 m - 16 - 1 \cdot m - 3 \cdot - 1)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Multiply $- 3$ by $- 1$ .

$\frac{m^{2} \cdot 4 (4 m - 16 - 1 \cdot m + 3)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Rewrite $- 1 m$ as $- m$ .

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

Subtract $m$ from $4 m$ .

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

Add $- 16$ and $3$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})}$

Step 5

Simplify the denominator.

Factor $m - 4$ out of $(m - 3) (m - 4) \cdot 16 + m^{2} (- {(m - 4)}^{2})$ .

Factor $m - 4$ out of $(m - 3) (m - 4) \cdot 16$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) ((m - 3) \cdot 16) + m^{2} (- {(m - 4)}^{2})}$

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

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) ((m - 3) \cdot 16) + (m - 4) (m^{2} (- (m - 4)))}$

Factor $m - 4$ out of $(m - 4) ((m - 3) \cdot 16) + (m - 4) (m^{2} (- (m - 4)))$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) ((m - 3) \cdot 16 + m^{2} (- (m - 4)))}$

Apply the distributive property.

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (m \cdot 16 - 3 \cdot 16 + m^{2} (- (m - 4)))}$

Move $16$ to the left of $m$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 \cdot m - 3 \cdot 16 + m^{2} (- (m - 4)))}$

Multiply $- 3$ by $16$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 \cdot m - 48 + m^{2} (- (m - 4)))}$

Rewrite using the commutative property of multiplication.

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - m^{2} (m - 4))}$

Apply the distributive property.

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - m^{2} m - m^{2} \cdot - 4)}$

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

Move $m$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - (m \cdot m^{2}) - m^{2} \cdot - 4)}$

Multiply $m$ by $m^{2}$ .

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

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - (m^{1} m^{2}) - m^{2} \cdot - 4)}$

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

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - m^{1 + 2} - m^{2} \cdot - 4)}$

Add $1$ and $2$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - m^{3} - m^{2} \cdot - 4)}$

Multiply $- 4$ by $- 1$ .

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (16 m - 48 - m^{3} + 4 m^{2})}$

Reorder terms.

$\frac{m^{2} \cdot 4 (3 m - 13)}{(m - 4) (- m^{3} + 4 m^{2} + 16 m - 48)}$

Step 6

Simplify with factoring out.

Move $4$ to the left of $m^{2}$ .

$\frac{4 \cdot m^{2} (3 m - 13)}{(m - 4) (- m^{3} + 4 m^{2} + 16 m - 48)}$

Factor $- 1$ out of $- m^{3}$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3}) + 4 m^{2} + 16 m - 48)}$

Factor $- 1$ out of $4 m^{2}$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3}) - (- 4 m^{2}) + 16 m - 48)}$

Factor $- 1$ out of $- (m^{3}) - (- 4 m^{2})$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3} - 4 m^{2}) + 16 m - 48)}$

Factor $- 1$ out of $16 m$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3} - 4 m^{2}) - (- 16 m) - 48)}$

Factor $- 1$ out of $- (m^{3} - 4 m^{2}) - (- 16 m)$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3} - 4 m^{2} - 16 m) - 48)}$

Rewrite $- 48$ as $- 1 (48)$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3} - 4 m^{2} - 16 m) - 1 (48))}$

Factor $- 1$ out of $- (m^{3} - 4 m^{2} - 16 m) - 1 (48)$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- (m^{3} - 4 m^{2} - 16 m + 48))}$

Rewrite negatives.

Rewrite $- (m^{3} - 4 m^{2} - 16 m + 48)$ as $- 1 (m^{3} - 4 m^{2} - 16 m + 48)$ .

$\frac{4 m^{2} (3 m - 13)}{(m - 4) (- 1 (m^{3} - 4 m^{2} - 16 m + 48))}$

Move the negative in front of the fraction.

$- \frac{4 m^{2} (3 m - 13)}{(m - 4) (m^{3} - 4 m^{2} - 16 m + 48)}$

Simplify (16/(m-3)-4/(m-4))/(16/(m^2)-(m-4)/(m-3))

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