№ 371 Алгебра = № 13.40 Математика
Обчисліть:
$
1) \ \frac{7^9 \cdot 49^8}{343^8}; \\
2) \ \frac{6^{12}}{2^{10} \cdot 3^{11}}; \\
3) \ \frac{2^8 \cdot 5^7}{100^3}; \\
4) \ \frac{36^5}{24^6}.
$
Розв'язок:
1) $\frac{7^9 \cdot 49^8}{343^8} = \frac{7^9 \cdot 7^{16}}{7^{24}} = \frac{7^{25}}{7^{24}} = 7$;
2) $\frac{6^{12}}{2^{10} \cdot 3^{11}} = \frac{6^{12}}{6^{10} \cdot 3} = \frac{6^2}{3} = \frac{36}{3} = 12$;
3) $\frac{2^8 \cdot 5^7}{100^3} = \frac{10^7 \cdot 2}{(10^2)^3} = \frac{10^7 \cdot 2}{10^6} = 10 \cdot 2 = 20$;
4) $\frac{36^5}{24^6} = \frac{(6^2)^5}{6^6 \cdot 4^6} = \frac{6^{10}}{6^6 \cdot 4^6} = \frac{6^4}{4^6} = \frac{6^4}{4^4 \cdot 4^2} =$
$= \left(\frac{6}{4}\right)^4 \cdot \frac{1}{16} = \left(\frac{3}{2}\right)^4 \cdot \frac{1}{16} = \frac{81}{16} \cdot \frac{1}{16} =$
$= \frac{81}{256}$.