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.

Solution