# How do you order the following from least to greatest without a calculator 8.1 times 10^3, 2.1times10^5, 7.25times10^3, 1.05times10^4, 1.75times10^4?

Jul 31, 2016

Use the power of 10 first, and then the coefficient

#### Explanation:

If we have powers of 10 as the ones given, the first observation is that:

if $i < j$ then ${10}^{i} < {10}^{j}$, as in ${10}^{3} < {10}^{5}$. See that this is simply saying $1 , 000 < 100 , 000$

The second observation is that if we have the same power ${10}^{j}$, and $a < b$, then $a \cdot {10}^{j} < b \cdot {10}^{j}$, as is $3 \cdot {10}^{2} < 7 \cdot {10}^{2}$

Now let's put them in order:

$7.25 \cdot {10}^{3} < 8.1 \cdot {10}^{3} < 1.05 \cdot {10}^{4} < 1.75 \cdot {10}^{4} < 2.1 \cdot {10}^{5}$