$a$)
Ta có:
$10^{30} = (10^3)^{10} = 1000^{10}$
$2^{100} = (2^{10})^{10} = 1024^{10}$
Vì : $1000^{10} < 1024^{10} ⇒ 10^{30} < 2^{100}$.
Ta có:
$3^{2n} = (3^2)^n = 9^n$
$2^{3n} = (2^3)^n = 8^n$
Vì : $9^n > 8^n ⇒ 3^{2n} > 2^{3n}$
$b$)
$M = 2^2 + 2^3 + 2^4 + ... + 2^{100}$
$⇔ 2M = 2.(2^2 + 2^3 + 2^4 + ... + 2^{100})$
$⇔ 2M = 2^3 + 2^4 + 2^5 + ... + 2^{101}$
$⇔ 2M - M= (2^3 + 2^4 + 2^5 + ... + 2^{101})-( 2^2 + 2^3 + 2^4 + ... + 2^{100})$
$⇔ M = 2^{101} - 2^2$
$⇔ M = 2^{101} - 4$.
