Упр.280 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$9^{2\log_3(5)}$$
2. $$\left(\frac{1}{9}\right)^{\frac{1}{2}\log_3(4)}$$
3. $$\left(\frac{1}{4}\right)^{-5\log_2(3)}$$
4. $$27^{-\log_{\frac{1}{3}}(5)}$$
5. $$10^{3-\log_{10}(5)}$$
6. $$\left(\frac{1}{7}\right)^{1+2\log_{\frac{1}{7}}(3)}$$

1. $$9^{2\log_3 5}=(3^2)^{2\log_3 5}=3^{4\log_3 5}=\left(3^{\log_3 5}\right)^4=5^4=625.$$

2. $$\left(\frac{1}{9}\right)^{\frac12\log_3 4}=\left(3^{-2}\right)^{\frac12\log_3 4}=3^{-\log_3 4}=\left(3^{\log_3 4}\right)^{-1}=4^{-1}=\frac14.$$

3. $$\left(\frac14\right)^{-5\log_2 3}=\left(2^{-2}\right)^{-5\log_2 3}=2^{10\log_2 3}=\left(2^{\log_2 3}\right)^{10}=3^{10}=59049.$$

4. $$27^{-4\log_{1/3} 5}=(3^3)^{-4\log_{1/3} 5}=3^{-12\log_{1/3} 5}=\left(\frac13\right)^{12\log_{1/3} 5}=\left(\left(\frac13\right)^{\log_{1/3} 5}\right)^{12}=5^{12}.$$

5. $$10^{3-\log_{10} 5}=\frac{10^3}{10^{\log_{10} 5}}=\frac{1000}{5}=200.$$

6. $$\left(\frac17\right)^{1+2\log_{1/7} 3}=\frac17\cdot \left(\frac17\right)^{2\log_{1/7} 3}=\frac17\cdot \left(\left(\frac17\right)^{\log_{1/7} 3}\right)^2=\frac17\cdot 3^2=\frac97=1\frac27.$$

Ответ: $$625;\ \frac14;\ 59049;\ 5^{12};\ 200;\ 1\frac27.$$