Lesson 2

Recall the basic properties of the exponentiation operation. For any positive real numbers \(a,b\) and real numbers \(p,q\), the following properties hold:

• \(a^{0}=1\)


• \((ab)^{p}=a^{p}b^{p}\)


• \(\left( \frac{a}{b}\right)^{p}=\frac{a^p}{b^p}\)


• \(a^{p}a^{q}=a^{p+q}\)


• \(\frac{a^p}{a^q}=a^{p-q}\)


• \(\left(a^{p}\right)^{q}=a^{pq}\)


• \(a^{-1}=\frac{1}{a}\)


• \(a^{-p}=\frac{1}{a^p}\)


• \(a^{\frac{1}{q}}=\sqrt[q]{a}\) for every positive integer \(q\)


• \(a^{\frac{m}{q}}=\sqrt[q]{a^m}=(\sqrt[q]{a})^{m}\) for every integer \(m\) and positive integer \(q\)

Problem. Calculate \[\frac{\left(18^3\right)^{2^3} \cdot 9^{-20}}{24^7}.\]