Question

1st attempt
solve the following unit conversion. pay attention to significant figures and the final unit.

( 12.7 \text{km} \times \frac{1000 \text{m}}{1 \text{km}} \times \frac{1000 \text{mm}}{1 \text{m}} = square \text{mm} )

Explanation:

Step1: Cancel km units

First, multiply \(12.7\space km\) by \(\frac{1000\space m}{1\space km}\). The \(km\) units cancel out, giving \(12.7\times1000\space m\).
\(12.7\times1000 = 12700\space m\)

Step2: Cancel m units

Next, multiply the result by \(\frac{1000\space mm}{1\space m}\). The \(m\) units cancel out, so we have \(12700\times1000\space mm\).
\(12700\times1000 = 12700000\space mm\)

Step3: Check significant figures

The original value \(12.7\space km\) has three significant figures. When multiplying by conversion factors (which are exact and don't affect significant figures), the result should also have three significant figures. In scientific notation, \(12700000\) with three significant figures is \(1.27\times10^{7}\), but in decimal form, we can write it as \(1.27\times10^{7}\) or \(12700000\) (noting that the trailing zeros are just placeholders here, but since we have three significant figures from \(12.7\), we present it as \(1.27\times10^{7}\) or \(12700000\) where the significant figures are 1, 2, 7). Wait, actually, when we do the multiplication: \(12.7\times1000\times1000 = 12.7\times10^{6}=1.27\times10^{7}\) (in scientific notation) or \(12700000\) (which is \(1.27\times10^{7}\) when considering significant figures). But let's do the calculation step by step with the numbers:

\(12.7\space km\times\frac{1000\space m}{1\space km}=12.7\times1000\space m = 12700\space m\)

Then \(12700\space m\times\frac{1000\space mm}{1\space m}=12700\times1000\space mm = 12700000\space mm\)

Since \(12.7\) has three significant figures, the result should be \(1.27\times10^{7}\space mm\) or \(12700000\space mm\) (the trailing zeros are not significant here, but the 1,2,7 are significant). Wait, actually, when multiplying, the number of significant figures is determined by the least number of significant figures in the factors. Here, \(12.7\) has three, and the conversion factors are exact (infinite significant figures), so the result should have three significant figures. So \(12.7\times1000\times1000 = 12.7\times10^{6}=1.27\times10^{7}\) (which is \(12700000\) when written out, and the first three digits 1,2,7 are significant).

Answer:

\(1.27\times10^{7}\) (or \(12700000\))