Simplifier l'écriture des nombres suivants

Question

  • \(2,7^{1,5} \times 2,7^{1,3}=\)

  • \(3^{2,4} \times 4^{2,4} =\)

  • \(\dfrac{1,1^{-0,7}}{1,1^{2,1}}=\)

  • \((8,3^{1,2})^3=\)

  • \(\dfrac{1,8^{-0,3}}{0,5^{-0,3}}=\)

  • \(\dfrac{1}{20,2^{0,5}}=\)

Solution

  • \(2,7^{1,5} \times 2,7^{1,3}=2,7^{1,5+1,3}=\color{red}2,7^{2,8}\)

    \(a^b\times a^d=\color{red}a^{b+d}\)

  • \(3^{2,4} \times 4^{2,4} =(3 \times 4)^{2,4}=\color{red}12^{2,4}\)

    \(a^b\times c^b=\color{red}(ac)^b\)

  • \(\dfrac{1,1^{-0,7}}{1,1^{2,1}}=1,1^{-0,7} \times 1,1^{-2,1}=1,1^{-0,7-2,1}={\color{red}1,1^{-2,8}}=\color{red}\dfrac{1}{1,1^{2,8}}\)

    \(\frac{a^b}{a^d}=\color{red}a^{b-d}-\)

  • \((8,3^{1,2})^3=8,3^{1,2 \times 3}=\color{red}8,3^{3,6}\)

    \((a^b)^d=\color{red}a^{b\times d}\)

  • \(\dfrac{1,8^{-0,3}}{0,5^{-0,3}}=(\frac{1,8}{0,5})^{-0,3}={\color{red}3,6^{-0,3}}={\color{red}(\frac{1}{3,6})^{0,3}}\)

    \(\frac{a^b}{c^b}=\color{red}(\frac{a}{c})^b\)

  • \(\dfrac{1}{20,2^{0,5}}=\frac{1}{\sqrt{20,2}}=\frac{1}{\sqrt{20,2}} \times \frac{\sqrt{20,2}}{\sqrt{20,2}}=\color{red}\frac{\sqrt{20,2}}{20,2}\)

    \(a^{\frac12}=\color{red}\sqrt{a}\)