Problem: Proving fraction rules

Problem

1. Prove fraction rule 4, $\frac{a}{b}⋅ \frac{c}{d}= \frac{a⋅c}{b⋅d}$ . Hint: You need $\frac{a}{b}=ab^{-1}$ , the commutative and associative rule and power rule 4.
2. Prove fraction rule 5, $\frac{a}{b}= \frac{a⋅c}{b⋅c}$ . Hint: You need fraction rule 3 and 4
3. Prove fraction rule 6, $\frac{a}{b}+ \frac{c}{b}= \frac{a+c}{b}$ . Hint: You need $\frac{a}{b}=ab^{-1}$ and the distributive law $a\left( b+c \right)=ab+ac$ .
4. Prove fraction rule 7 and 8, $\frac{-a}{b}= \frac{a}{-b}=- \frac{a}{b}$ and $\frac{-a}{-b}= \frac{a}{b}$ . Hint: $-a=\left( -1 \right)⋅a$ plus fraction rule 4.
5. Prove that ${\left( \frac{a}{b} \right)}^{-1}= \frac{b}{a}$ . Hint: Fraction rule 3, first part.
6. Prove fraction rule 9, $\frac{a/b}{c/d}= \frac{a}{b}⋅ \frac{d}{c}= \frac{ad}{bc}$ . Hint: part e)
7. Prove fraction rule 10, $\frac{a}{b}+ \frac{c}{d}= \frac{ad+bc}{bd}$ . Hint: You need fraction rule 5 and 6.