Question

Simplify each expression. Write your answer in exponential notation. PLEASE HELP

Answer 1

$\bf ~~~~~~~~~~~~\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} \qquad \qquad \cfrac{1}{a^n}\implies a^{-n} \qquad \qquad a^n\implies \cfrac{1}{a^{-n}} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{25^2\cdot 25^6}{5^4\cdot 5}\implies \cfrac{(5^2)^2\cdot (5^2)^6}{5^{4+1}}\implies \cfrac{5^{2\cdot 2}\cdot 5^{2\cdot 6}}{5^5}\implies \cfrac{5^4\cdot 5^{12}}{5^5} \\\\\\ \cfrac{5^{4+12}}{5^5}\implies \cfrac{5^{16}}{5^5}\implies 5^{16}\cdot 5^{-5}\implies 5^{16-5}\implies 5^{11}\implies 48828125$

$\bf ~\dotfill\\\\ \cfrac{8^7\cdot 8^4\cdot 2^3}{16^3}\implies \cfrac{(2^3)^7\cdot (2^3)^4\cdot 2^3}{(2^4)^3}\implies \cfrac{2^{21}\cdot 2^{12}\cdot 2^3}{2^{12}}\implies 2^{21}\cdot 2^{12}\cdot 2^3\cdot 2^{-12} \\\\\\ 2^{21+12+3-12}\implies 2^{24}\implies 16777216 \\\\[-0.35em] ~\dotfill\\\\ \cfrac{18^8}{6^3\cdot 3^4\cdot 6^5}\implies \cfrac{(2\cdot 3^2)^8}{6^{3+5}\cdot 3^4}\implies \cfrac{2^8\cdot 3^{2\cdot 8}}{6^8\cdot 3^4}\implies \cfrac{2^8\cdot 3^{16}}{(2\cdot 3)^8\cdot 3^4}$

$\bf \cfrac{2^8\cdot 3^{16}}{2^8\cdot 3^8\cdot 3^4}\implies \cfrac{2^8\cdot 3^{16}}{2^8\cdot 3^{8+4}}\implies \cfrac{2^8\cdot 3^{16}}{2^8\cdot 3^{12}}\implies \cfrac{2^8}{2^8}\cdot \cfrac{3^{16}}{3^{12}}\implies 1\cdot 3^{16}\cdot 3^{-12} \\\\\\ 3^4\implies 81$