Real Numbers

7/16, 13/125, 17/20

For terminating decimals, denominator must have only factors 2 and 5. 16 = 2⁴ ✓, 125 = 5³ ✓, 20 = 2² × 5 ✓. But 18 = 2 × 3² (has factor 3) ✗, 30 = 2 × 3 × 5 (has factor 3) ✗.

Assume √3 is rational, then 3 divides both numerator and denominator

Assume √3 = p/q where p,q are coprime. Then 3q² = p², so 3 divides p². By theorem, 3 divides p. Let p = 3k, then 3q² = 9k², so q² = 3k², meaning 3 divides q. This contradicts p,q being coprime.

2 × 3 × 5 × 7 × 11

2310 ÷ 2 = 1155, 1155 ÷ 3 = 385, 385 ÷ 5 = 77, 77 ÷ 7 = 11, 11 ÷ 11 = 1. So 2310 = 2 × 3 × 5 × 7 × 11.

log_a(xy) = log_a(x) + log_a(y), log_a(x/y) = log_a(x) - log_a(y), log_a(x^n) = n × log_a(x), log_a(a) = 1

These are the fundamental properties: product rule, quotient rule, power rule, and log of base equals 1. The option log_a(x + y) = log_a(x) × log_a(y) is incorrect.