BConsider the number of digits you must write when writing out small numbers longhand instead of as negative powers of 10.
A3 * 10^(-7)
BSee solution.
Practice makes perfect
We are asked to express 0.000000298 as a single digit times a power of ten rounded to the nearest ten millionth. Ten million has 7 zeros, which means that we want to keep 7 zeros and then have one non-zero digit in our number. Because we have 9 after 2 in our number, we will round it to 0.0000003.
0.000000298 ≈ 0.0000003
Now, we want to rewrite this number as a single digit times a power of 10. We already know that there are 7 zeros. Our number is less than 1, so the exponent of 10 is negative.
0.000000298 ≈ 0.0000003 = 3 * 10^(-7)
Now we are asked to explain how negative powers of 10 make small numbers easier to write and compare. To do so, let's take a look at two small numbers 0.000000000723 and 0.0000000431. We cannot immediately say how many zeros there are, which also makes it hard to compare them. Let's rewrite them as a single digit times a power of 10.
0.000000000723 ≈ 0.0000000007 = 7 * 10^(-10)
0.0000000431 ≈ 0.00000004 = 4 * 10^(-8)
Now that these numbers are written with negative powers of 10, we can tell that 7 * 10^(-10) is less than 4 * 10^(-8), because 10^(-10) is less than 10^8. Also, the numbers are easier to write in this form because we do not need to write all the zeros.
