Question

Comparati:
a) 5^8 cu 2^15+2×4^7
b) 5^14 cu 3^21​

Answer 1

a) 5⁸ cu 2¹⁵ + 2 × 4⁷

$\bf 2^{15}+ 2 \cdot4^7 =2^{15} + 2 \cdot( 2^{2}) ^7 =$

$\bf 2^{15} + 2 \cdot2^{2 \cdot7} = 2^{15} + 2 \cdot2^{14} =$

$\bf 2^{15} + 2^{1 + 14} = 2^{15} + 2^{15} =$

$\bf 2^{15} \cdot( 2^{15 - 15} +2^{15 - 15}) =$

$\bf 2^{15} \cdot( 2^{0} +2^{0}) =2^{15} \cdot( 1+1) =$

$\bf 2^{15} \cdot2 =2^{15 + 1} = 2^{16} =$

$\bf (2^{2})^{8} =\purple{ \underline{4^{8}}}$

$\bf 5^{8} > 4^{8} \implies \red{ \underline{ \: 5^{8} \: > \: 2^{5} + 2 \cdot 4^{7} \: }}$

______________________

b) 5¹⁴ cu 3²¹

$\bf 5^{14} = 5^{2 \cdot7} = (5^2)^7 = \purple{ \underline{25^7}}$

$\bf 3^{21} = 3^{3 \cdot7} = (3^3)^7 = \pink{ \underline{27^7}}$

$\bf 25^{7} < 24^{7} \implies \red{ \underline{ \: 5^{14} \: < \: 3^{21} \: }}$