Question

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Answer 1

\(3.\)

\(a.\dfrac{8^8}{2^7\cdot4^9}=\dfrac{\left(2^3\right)^8}{2^7\cdot\left(2^2\right)^9}=\dfrac{2^{24}}{2^{25}}=\dfrac{1}{2}\)

\(\dfrac{6^8\cdot12^8}{8^9\cdot9^9}=\dfrac{\left(2\cdot3\right)^8\cdot\left(2^2\cdot3\right)^8}{\left(2^3\right)^9\cdot\left(3^2\right)^9}=\dfrac{2^{24}\cdot3^{16}}{2^{27}\cdot3^{18}}=\dfrac{1}{2^3\cdot3^2}\)

\(\dfrac{45^{10}\cdot5^{10}}{75^{10}}=\dfrac{\left(5\cdot3^2\right)^{10}\cdot5^{10}}{\left(5^2\cdot3\right)^{10}}=\dfrac{5^{20}\cdot3^{20}}{5^{20}\cdot3^{10}}=3^{10}\)

\(\dfrac{2^{15}\cdot9^4}{6^6\cdot8^3}=\dfrac{2^{15}\cdot\left(3^2\right)^4}{\left(3\cdot2\right)^6\cdot\left(2^3\right)^3}=\dfrac{2^{15}\cdot3^8}{3^6\cdot2^{15}}=3^2\)

Answer 2

a) \(\dfrac{8^8}{2^7\cdot4^9}=\dfrac{2^{24}}{2^7\cdot2^{18}}=\dfrac{1}{2}\)

b) \(\dfrac{6^8\cdot12^8}{8^9\cdot9^9}=\dfrac{2^8\cdot3^8\cdot2^{16}\cdot3^8}{2^{27}\cdot3^{18}}=\dfrac{1}{8}\cdot\dfrac{1}{9}=\dfrac{1}{72}\)

c) \(\dfrac{45^{10}\cdot5^{10}}{75^{10}}=\left(\dfrac{225}{75}\right)^{10}=3^{10}\)

d) \(\dfrac{2^{15}\cdot9^4}{6^6\cdot8^3}=\dfrac{2^{15}\cdot3^8}{2^6\cdot3^6\cdot2^9}=3^2=9\)