Exercice 00-05
Simplifier les expressions, où \(a,b>0\) et \(p,q\in \mathbb{R}^*\).
  1. \((ab)^{p}b^{q-p}\)
  2. \(\frac{a^{p}}{b^{-q}}\)
  3. \(\frac{b^{q}}{a^{-p}}\)
  4. \((ab^{\frac{q}{p}})^{p}\)
  5. \((a^{\frac{p}{q}}b)^{q}\)
  6. \(\bigl(a^{\frac{1}{q}}b^{\frac{1}{p}}\bigr)^{pq}\)
  7. \(\sqrt{a^{2p}}\,b^{q}\)
  8. \(\bigl((\frac{1}{a})^q + (\frac{1}{b})^p\bigr) \frac{a^p(ab)^q}{1+\frac{a^q}{b^p}}\)
  9. \(a^q b^p\,\frac{\frac{a^p + b^q}{(\frac{1}{a})^{p} +(\frac{1}{b})^{q}}}{\frac{a^{q}+b^{p}}{(\frac{1}{b})^{p}+(\frac{1}{a})^{q}}}\)
  10. \(a^q b^p\!((a^{\frac{1}{q}-\frac{1}{p}}\, b^{\frac{1}{p}-\frac{1}{q}})^{\!p\:})^{\!q}\)
Chacune de ces expressions est égale à \(a^pb^q\).