Error in my textbook? scientific notation conversion

[ikihi]

Homework Statement

The question is: The Earth formed 4.57 x 10^9 years ago. What is this time in seconds?

The answer the textbook is giving is 1.44x10^17 seconds. The answer I keep getting is 1.44x10^15 seconds. Is the textbook wrong or am i wrong?

 

[gneill]

Homework Statement

The question is: The Earth formed 4.57 x 10^9. What is this time in seconds?

The answer the textbook is giving is 1.44x10^17 seconds. The answer I keep getting is 1.44x10^15 seconds. Is the textbook wrong or am i wrong?

Looks like the textbook is correct. Show your work.

 

[ikihi]

Looks like the textbook is correct. Show your work.

Whoops! I was multiplying 4.57x10^7 instead of the given 4.57x10^9. I think I got it now.

I was wondering though. What is the correct way of setting up this problem. I went ahead and expanded 4.57x10^9 years into 4,570,000,000 years.

Next I did unit conversion to go from years to seconds. I got 525,600 minutes in 1 year and 60 seconds in one minute. Then I multiplied 4,570,000,000 and 525,600 and 60 to get the full time in seconds. The answer comes to 1.44x10^17 rounded for 3 sig figs.

However I noticed that I could convert back to scientific notation and get the same answer. I changed 525,600 min to 5.256x10^5 min and 60 seconds to 6x10^1 seconds.

Then I multiplied together (4.57x10^9) years (5.256x10^5) min. (6x10^1)seconds
=(4.57x10^9) x (5.257x10^5) x (6x10^1) = 1.44x10^17 (the same answer)

My question is that is it necessary to change into those large numbers, or is there an easier way to do it.

 

[gneill]

You can leave everything in scientific notation or mix and match as desired. A change of notation doesn't change the value of a number.

You may find it helpful to set up the sequence of conversions as a series of multiplications by 1:

$$4.57x10^7\;yr \times \frac{365.25 days}{yr} \times \frac{24hr}{day} \times \frac{60 min}{hr} \times ...$$
Note that each fraction is equivalent to unity, with the numerator and denominator being equal to the same thing in different units. Note how the units cancel as the sequence progresses, eventually leaving you with just the units you were after.

If your calculator is in scientific notation mode, multiplying a number in scientific notation by another number not in scientific notation will return a result in scientific notation. It's often easier to carry around numbers in that format, particularly when only a few significant digits are required.

 

[ikihi]

You can leave everything in scientific notation or mix and match as desired. A change of notation doesn't change the value of a number.

You may find it helpful to set up the sequence of conversions as a series of multiplications by 1:

$$4.57x10^7\;yr \times \frac{365.25 days}{yr} \times \frac{24hr}{day} \times \frac{60 min}{hr} \times ...$$
Note that each fraction is equivalent to unity, with the numerator and denominator being equal to the same thing in different units. Note how the units cancel as the sequence progresses, eventually leaving you with just the units you were after.

If your calculator is in scientific notation mode, multiplying a number in scientific notation by another number not in scientific notation will return a result in scientific notation. It's often easier to carry around numbers in that format, particularly when only a few significant digits are required.

Hmm I see. I think my problem was that I skipped steps on the conversion part from years to seconds and got confused. I went from years to minutes instead of going in order from years to days to hours to seconds.

And I didn't realize I could just enter that into my calculator and I would get the answer that quick. (4.57X10^9)(365.25)(24)(60)(60)= 1.44x10^17 seconds.

Thankyou.

 

[litup]

Also, one day = 86400 seconds, so if you remember that, just multiply it times 365.25.