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

The Correct Option is B

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 \(b\).