Упр.76 ГДЗ Алимов 10-11 класс (Алгебра)

Вычислить:

1. $$\left(\frac{1}{16}\right)^{-0{,}75}+810000^{0{,}25}-\left(7\frac{19}{32}\right)^{\frac{1}{5}}$$;
2. $$27^{\frac{2}{3}}-(-2)^{-2}+\left(3\frac{3}{8}\right)^{-\frac{1}{3}}$$;
3. $$\left(0{,}001\right)^{-\frac{1}{3}}-2^{-2}\cdot64^{\frac{2}{3}}-8^{-1\frac{1}{3}}$$;
4. $$(-0{,}5)^{-4}-625^{0{,}25}-\left(2\frac{1}{4}\right)^{-1\frac{1}{2}}$$.

1)

$$\left(\frac{1}{16}\right)^{-0,75}+810000^{0,25}-\left(7\frac{19}{32}\right)^{\frac15}$$
$$=\left(\frac{1}{16}\right)^{-\frac34}+\left(810000\right)^{\frac14}-\left(\frac{243}{32}\right)^{\frac15}$$
$$=(2^{-4})^{-\frac34}+(3^4\cdot 10^4)^{\frac14}-\left(\frac{3^5}{2^5}\right)^{\frac15}$$
$$=2^3+3\cdot 10-\frac32=8+30-1,5=36,5.$$

Ответ: $$36,5$$

2)

$$27^{\frac23}-(-2)^{-2}+\left(3\frac38\right)^{-\frac13}$$
$$=(3^3)^{\frac23}-\left(-\frac12\right)^{-2}+\left(\frac{27}{8}\right)^{-\frac13}$$
$$=3^2-\frac14+\left(\frac{8}{27}\right)^{\frac13}$$
$$=9-\frac14+\frac23=\frac{113}{12}=9\frac{5}{12}.$$

Ответ: $$9\frac{5}{12}$$

3)

$$\left(0,001\right)^{-\frac13}-2^{-2}\cdot 64^{\frac23}-8^{-1\frac13}$$
$$=\left(10^{-3}\right)^{-\frac13}-2^{-2}\cdot (2^6)^{\frac23}-8^{-\frac43}$$
$$=10- \frac14\cdot 2^4-\left(2^3\right)^{-\frac43}$$
$$=10-4-\frac{1}{2^4}=10-4-\frac{1}{16}=5\frac{15}{16}.$$

Ответ: $$5\frac{15}{16}$$

4)

$$(-0,5)^{-4}-625^{0,25}-\left(2\frac14\right)^{-1\frac12}$$
$$=\left(-\frac12\right)^{-4}-\left(5^4\right)^{\frac14}-\left(\frac94\right)^{-\frac32}$$
$$=16-5-\left(\frac49\right)^{\frac32}$$
$$=11-\left(\frac{2^2}{3^2}\right)^{\frac32}=11-\frac{2^3}{3^3}=11-\frac{8}{27}=10\frac{19}{27}.$$

Ответ: $$10\frac{19}{27}$$