Number Theory and Computation

$18 = 2 \times 3^2$ and $30 = 2 \times 3 \times 5$. The shared primes at their lowest powers are $2 \times 3 = 6$.

Multiplication comes before subtraction: $3 \times 4 = 12$, then $20 - 12 = 8$.

$12 = 2^2 \times 3$ and $18 = 2 \times 3^2$. Taking every prime at its highest power gives $2^2 \times 3^2 = 36$.

Put the point after the 4 to get $4.58$. The point has moved 4 places to the left, and the number is larger than 1, so the index is $+4$.

$\dfrac23 + \dfrac14 = \dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$. Dividing means multiplying by the reciprocal: $\dfrac{11}{12} \times \dfrac{6}{11} = \dfrac{6}{12} = \dfrac12$.

The point moves 3 places right to sit after the 3, and the number is less than 1, so the index is $-3$. The leading zeros are placeholders, but the zero between 3 and 5 is significant, giving 3 significant figures.