Fraction Division B14Dd1

1. **State the problem:** Simplify the expression $$\frac{mk-2k}{3k} \div \frac{m^2 - 4}{6}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{mk-2k}{3k} \div \frac{m^2 - 4}{6} = \frac{mk-2k}{3k} \times \frac{6}{m^2 - 4}$$. 3. **Factor where possible:** - Factor numerator $$mk - 2k = k(m - 2)$$. - Recognize $$m^2 - 4$$ as a difference of squares: $$m^2 - 4 = (m - 2)(m + 2)$$. 4. **Rewrite the expression:** $$\frac{k(m - 2)}{3k} \times \frac{6}{(m - 2)(m + 2)}$$. 5. **Cancel common factors:** - Cancel $$k$$ in numerator and denominator: $$\frac{\cancel{k}(m - 2)}{3\cancel{k}} \times \frac{6}{(m - 2)(m + 2)} = \frac{m - 2}{3} \times \frac{6}{(m - 2)(m + 2)}$$. - Cancel $$m - 2$$ in numerator and denominator: $$\frac{\cancel{m - 2}}{3} \times \frac{6}{\cancel{m - 2}(m + 2)} = \frac{1}{3} \times \frac{6}{m + 2}$$. 6. **Multiply the remaining fractions:** $$\frac{1 \times 6}{3 \times (m + 2)} = \frac{6}{3(m + 2)}$$. 7. **Simplify the fraction $$\frac{6}{3(m + 2)}$$:** $$\frac{\cancel{6}^2}{\cancel{3}^1 (m + 2)} = \frac{2}{m + 2}$$. **Final answer:** $$\boxed{\frac{2}{m + 2}}$$.