### Video Transcript

Which of the following is the smallest? Is it (A) 0.4 times 10 to the power of six, (B) 0.7 times 10 to the power of eight, (C) 51.2 times 10 to the power of eight, or (D) 51.2 times 10 to the power of six?

If we look at these four numbers, we notice that they are all written in a form similar to scientific notation, since they are each a number times a power of 10.

Recall that a number written in scientific notation is of the form 𝑎 times 10 to the power of 𝑛, where the modulus of 𝑎, which is just the size of the number, has to be between one and 10. As a matter of fact, none of these numbers have a modulus in this range. So they are not technically in standard form. In order to compare the numbers, it is more convenient if we convert them to scientific notation. This is because if we have two numbers in scientific form and one has a higher exponent of 10 than the other, then it will be larger. Also, if two numbers in scientific notation have the same powers of 10, then the one with the bigger multiplier is bigger.

Hence, let us convert our four options to scientific form, starting with 0.4 times 10 to the six. This is the same as 0.4 times 10 times 10 to the six divided by 10, which simplifies to four times 10 to the power of five. In much the same way, we can rewrite 0.7 times 10 to the eight as seven times 10 to the seven.

Now, so far, the power of 10 has decreased by one in each case after we have converted the number to scientific notation. But this is because the multipliers were less than one. For options (C) and (D), the multipliers are greater than 10. So we can expect the exponent of 10 to increase when we convert them. For 51.2 times 10 to the eight, we can divide 51.2 by 10 and multiply the 10s by 10, to find that this is equal to 5.12 times 10 to the nine. And similarly, option (D) is equivalent to 5.12 times 10 to the seven.

So, let us compare these four numbers. If we use the first comparison property here, we can immediately tell that four times 10 to the five has the smallest power of 10. So it is the smallest number. In fact, we can order all the numbers as shown. In particular, the two quantities in the middle can be ordered using the second comparison quality, since 5.12 is less than seven. Therefore, the smallest number is option (A), which was originally 0.4 times 10 to the six. And we’ve now converted it to four times 10 to the power of five.