In an expression \(a \times 10^b\):
- \(a\) is the order of magnitude for \(b \leq 5\)
- \(b\) is the order of magnitude for \(a \leq 5\)
- \(b\) is the order of magnitude for \(5<a \leq 10\)
- \(b\) is the order of magnitude for \(a \geq 5\)
The Correct Option is B
Approach Solution - 1
To understand the expression \(a \times 10^b\), we need to determine what the "order of magnitude" refers to, especially in the context of scientific notation.
In scientific notation, any number can be expressed as \(a \times 10^b\), where:
- \(a\) is the coefficient which must be greater than or equal to 1 and less than 10, i.e., \(1 \leq a < 10\).
- \(b\) is the exponent of 10 and determines the order of magnitude of the number.
The order of magnitude refers specifically to the power of 10 that best fits the scale of the number. In other words, it is the exponent \(b\) in the expression.
Exploring the given options:
- \(a\) is the order of magnitude for \(b \leq 5\): This is incorrect since \(a\) is simply the coefficient and not related to the order of magnitude based on \(b\).
- \(b\) is the order of magnitude for \(a \leq 5\): This is correct because \(b\), the exponent, represents the order of magnitude of the number irrespective of the value of \(a\) as long as \(1 \leq a < 10\).
- \(b\) is the order of magnitude for \(5 < a \leq 10\): This is misleading and incorrect. The value of \(a\) has no effect on the definition of the order of magnitude, which is strictly determined by \(b\).
- \(b\) is the order of magnitude for \(a \geq 5\): This statement is incorrect as it suggests a dependency on \(a\), which does not determine the order of magnitude.
Hence, the correct answer is: \(b\) is the order of magnitude for \(a \leq 5\).
This conclusion holds because regardless of \(a\)'s specific value within its allowed range, it is the exponent \(b\) in \(10^b\) that dictates the order of magnitude of the expression.
Approach Solution -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 \(b\).
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