Video: Converting between Scientific Notation and Decimal Notation

A bullet comes to rest in a time of 5 × 10⁻⁴ s. What is the time taken for the bullet to come to rest, expressed in decimal form?

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Video Transcript

A bullet comes to rest in a time of five times 10 to the power of negative four seconds. What is the time taken for the bullet to come to rest, expressed in decimal form?

Okay, so in this question, what we’re being asked to do is to express this time, the time taken for the bullet to come to rest, in decimal form. When currently it’s been given to us in scientific notation. Let’s recall that scientific notation, sometimes known as standard form, is a very particular way to display a numerical value. A number is written in the form 𝑎 times 10 to the power of 𝑏, where 𝑎 is a number that’s greater than or equal to one and less than 10. And 𝑏 is an integer or a whole number. So any number written in this form, where we’ve got some number between one and 10 multiplied by 10 to the power of some integer, some whole number. Then we say that it’s written in scientific notation. And that is exactly what we have here. We’ve got five times 10 to the power of negative four.

In this particular case, our value of five is 𝑎 because it’s greater than or equal to one and less than 10. And negative four is our value of 𝑏, because it’s an integer. So we need to take this value, five times 10 to the power of negative four, and write it in decimal form, which is where we display the number in full. And as well as this, we need to remember to include the unit, seconds, because we’ve been asked to find the time taken for the bullet to come to rest expressed in decimal form.

So the value that we’ve been given for the time taken for the bullet to come to rest is five times 10 to the power of negative four seconds. But then we can recall that 10 to the power of negative four is the same thing as one divided by 10 to the power of four. This is what a negative power means. If we’ve got something to a negative power, then that’s the same thing as one divided by that thing to that positive power. And then, we can recall that 10 to the power of four is the same thing as 10 times 10 times 10 times 10. In other words, multiplying 10 four times. And so we can replace 10 to the power four by 10 times 10 times 10 times 10. And we can simplify the denominator because 10 times 10 is 100 times 10 is 1000 times 10 is 10000. And so what we’ve got now is that this fraction is equivalent to one divided by 10000.

But remember, one divided by 10000 is equivalent to 10 to the power of negative four. And so we can replace 10 to the power of negative four by one divided by 10000. And so what we’re left with now is five multiplied by one divided by 10000 seconds. And then we can simplify five multiplied by one divided by 10000. Because if we multiply that five by the numerator, we get five divided by 10000. And then we can simplify five divided by 10000 because that is equal to 0.0005 seconds. So that’s three zeros after the decimal point and then five. At which point, we’ve displayed the number in full. And therefore we’ve written it in decimal form.

Therefore, our final answer is that the amount of time taken for the bullet to come to rest, expressed in decimal form, is 0.0005 seconds.

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