Question
\( \begin{array}{lll} & & 3,458.9 \div \square \\ \text { A. } 10^{1} \\ \text { B. } 10^{2} \\ \text { C. } 10^{3} \\ \text { D. } 10^{4}\end{array} \)
Ask by Beck Jimenez. in the United States
Mar 18,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The correct choice is \(10^3\).
Solution
\[
\text{Step 1: Identify the problem.}
\]
We have the number \(3,458.9\) and we want to choose a power of 10 from the options \(10^1,\,10^2,\,10^3,\,10^4\) such that dividing \(3,458.9\) by that power of 10 gives a result in the range from 1 (inclusive) to 10 (exclusive). This format is characteristic of scientific notation.
\[
\text{Step 2: Write } 3,458.9 \text{ in scientific notation.}
\]
To do that, we want to express \(3,458.9\) as \(a \times 10^n\) where \(1 \leq a < 10\).
Observe that:
\[
3,458.9 = 3.4589 \times 10^3.
\]
\[
\text{Step 3: Interpret the division.}
\]
Since
\[
\frac{3,458.9}{10^3} = 3.4589,
\]
dividing by \(10^3\) converts the original number into its scientific notation form (with the coefficient \(a = 3.4589\) in the proper range).
\[
\text{Step 4: Choose the correct option.}
\]
Among the given options, the power of 10 that works is \(10^3\).
Thus, the answer is:
\[
\boxed{10^3}
\]
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Extra Insights
To solve \(3,458.9 \div \square\), we need to find out which power of ten would result in a quotient that is close to 1, as that will help us determine the correct option. Dividing \(3,458.9\) by \(10^3\) (or \(1,000\)) gives us approximately \(3.4589\). Dividing by \(10^2\) (or \(100\)) gives \(34.589\), which is too high, and dividing by \(10^4\) (or \(10,000\)) yields \(0.34589\), which is too low. Therefore, the only option that keeps the quotient above 1 but reasonably close to it is \(10^3\). So, the answer is \( \text{C. } 10^{3} \).