a. \( 23^{18} \cdot\left(23^{5}\right)^{4}:\left(23^{30} \cdot 23^{7}\right) \cdot 23= \) b. \( 6^{58}:\left(6^{13}\right)^{3}: 6^{19} \cdot\left(6^{2}\right)^{2}= \) c. \( \left(10^{15}\right)^{10} \cdot 10^{124}:\left(10^{10}\right)^{20}: 10^{70}= \) d. \( \sqrt[5]{5} \cdot \sqrt[5]{5^{3}} \cdot \sqrt[5]{5^{4}}: \sqrt[5]{5^{3}}= \) e. \( \sqrt[6]{16} \cdot \sqrt[6]{8}: \sqrt[6]{2}= \) f. \( \sqrt{81 \cdot 64: 144}= \)

Let’s explore how exponents and roots play a significant role in mathematics! When you're dealing with exponents \( (a^m \cdot a^n = a^{m+n}) \) and roots \( \sqrt[n]{a^m} = a^{m/n} \), you can simplify expressions dramatically. These properties help in tackling complex equations with ease, ensuring you never feel out of your depth in algebra! Now, wouldn’t it be interesting to see how these concepts apply in the real world? Engineers and scientists frequently use exponentiation for calculations involving growth rates, like population or investment growth, and in constructing models for physical phenomena. Understanding this can help learners appreciate the practical side of exponentiation beyond just academic exercises!