Express each of the following numbers in standard scientific notation. a. 0.5012 b. 5,012,000 C.

Express each of the following numbers in standard scientific...

Key Concepts

Scientific Notation

Scientific notation is a method for expressing very large or very small numbers in a compact form by representing them as the product of a number (the coefficient) between 1 and 10 and a power of 10. This notation simplifies calculations and the understanding of the scale of the number.

Coefficient and Exponent

In scientific notation, the number is split into a coefficient and an exponent. The coefficient is a decimal number that must be at least 1 but less than 10, and the exponent indicates how many places the decimal point was moved. A positive exponent signifies that the decimal was moved to the left (for large numbers), while a negative exponent indicates that it was moved to the right (for small numbers).

Decimal Point Movement

The process of converting a number to scientific notation involves shifting the decimal point so that the remaining number (the coefficient) falls between 1 and 10. The number of places the decimal point is moved becomes the exponent of 10, with the direction of the movement determining whether the exponent is positive or negative.

Transcript

00:02 Scientific notation is a way to rewrite a number to make it easier to read.

00:06 The format of scientific notation always is the same where you have a number to the left of the decimal point, and then after the decimal point you have three numbers to the thousandth decimal place.

00:20 The number to the left of the decimal can't be a zero, and following the number, you can have an exponent that shows what factor of 10 you change the original number by.

00:35 Some rules for when you are doing scientific notation is that if the decimal point is moved to the left, the exponent is going to be a negative number, and if the decimal point is moved to the right, the exponent is going to be a positive number.

00:47 If you don't need to move the decimal point at all, because the number is already in proper scientific format, then you don't even need to use an exponent.

00:56 So starting with the first number, we have 0 .5012.

01:02 In order to follow the format that we talked about above, we're going to want to move this decimal point, one spot, to the right.

01:11 If we rewrite that, now it's 5 .012.

01:16 Like we said in our rules, you're going to multiply that number by a factor of 10.

01:23 Because we move the decimal point to the right, the exponent is going to be a negative number, and because we moved it over one spot, it's going to be negative 1.

01:33 So the final answer is 5 .012 times 10 to the negative 1.

01:38 Look at the next question.

01:42 Our number is 5 ,012 ,000.

01:45 This is a really big number, so we're going to want to write it in a way that is smaller and easier to read.

01:51 We don't see any decimal points in here, which means there is an implied decimal point all the way at the end of this number, followed by some zeros.

02:03 Now we want to move the decimal point over to the left to be in the format that we talked about before.

02:09 So we do that together, removing the decimal point over 1, two, three, four, five, six spots.

02:20 And that's going to get us to 5 .012, which is the correct format with three numbers after the decimal.

02:28 And we're gonna add our exponent.

02:31 So we move this to the left.

02:34 So our number here is going to be positive.

02:36 We don't need to write positive.

02:39 If there is no negative sign next to a number, we assume is a positive number.

02:42 So this is going to be a six up here because we move the decimal point over six spots.

02:48 So our final answer is 5 .012 times 10 to the 6th.

02:55 Our next number is a really, really small number.

02:58 Our decimal point is way over here, and our number that's not a zero doesn't start until way over here...

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