Question

In an expression $a \times 10^b$:

• A. $a$ is the order of magnitude for $b \leq 5$
• B. $b$ is the order of magnitude for $a \leq 5$
• C. $b$ is the order of magnitude for $5<a \leq 10$
• D. $b$ is the order of magnitude for $a \geq 5$

Answer 1

Answer: (B)

$b$ is the order of magnitude for $a \leq 5$

Answer 2

The expression $a \times 10^b$ is in scientific notation, where:
$1 \leq a < 10, \quad \text{and } b \text{ is an integer}.$
The order of magnitude of a number is the power of 10 closest to that number. The value of $a$ determines how the exponent $b$ is interpreted:
Case 1: $a \leq 5$ When $a$ is less than or equal to 5, the number is closer to $10^b$ than $10^{b+1}$.
Therefore, the order of magnitude is:
$\text{Order of magnitude} = b.$
Case 2: $a > 5$ When $a$ is greater than 5, the number is closer to $10^{b+1}$ than $10^b$. In this case, the order of magnitude becomes:
$\text{Order of magnitude} = b + 1.$
The problem specifies $a \leq 5$, so the order of magnitude directly matches the exponent $b$.