Algebra II: Scientific Notion

Question

(0.00057) (77,426x 10³)

(38,000,000)(0.000473x10^-4)

^{how would you solve this and get the answer in scientific notion?}

Answer 1

Hello, Sally,

There are a variety of techniques for solving this mess of numbers. Here is how I approach it and why it makes it much easier, but also helps prevent mistakes (I hope - I'm sleepy at the moment, so check carefully).

The first step I take is to convert everything into scientific notation. Take the problem

(0.00057) (77,426x 10³)

(38,000,000)(0.000473x10^-4)

Every number is converted as below. Note how the exponent correlates with the number of digits I move the decimal point. Every step to the right adds a -1 to the exponent. Every digit to the left adds +1 to the exponent.

0.00057 = 5.7 x 10^-4

77.426 x 103 = 7.7426 x 10⁴ NOTE: This should have been 77,426, not 77.426. See the comments below and change this to 7.7426 x 10⁷. Please reflect that in the rest of the calculations. Sorry, Bob.

38,000,000 = 3.8 x 10⁷

0.000473 x 10-4 = 4.73 x 10^-8

With everything in scientific notation, we can separate the numbers from the collection of exponents and calculate them. While still busy, I've grouped them below, and it greatly simplifies calculator entry:

(5.7x7.7426)/(3.8x4.73)

The nice feature here is that I can estimate in my head what this portion of the number might be. I mentally do this calculation in this manner:

6x8 = 48 on top and 4x5 = 20 on bottom. This portion ought to be around 50/20 or 2 1/2 and so I'm aware if I get something entirely different I may have entered something incorrectly.

The best part is dealing with all of the 10^x units.

Arrange them the way they appear in the problem:

The top is 10^-4 x10⁴

The bottom is 10⁷ x10^-8

The exponents are added when multiplying.

For the top number = 10⁰ , which is 1.

The bottom number becomes 10^-1

We now have 10⁰/10^-1

When dividing, the exponent on the bottom is subtracted from the one on top.

10⁰/10^-1 = 10¹ which is simply 10.

I calculated 2.455 for the set of numbers and the power of 10 is 1, as we just discovered. My calculated value (2.455) is very close to what I mentally estimated (2.5), so I'm satisfied I entered the data reasonably correctly. All those exponents were reduced to 0, and now recombine the two portions of the calculation to get

2.455 x 10¹ or 24.55

I hope this helps. Keeping the exponent portion of numbers separate from the actual values make it easier, and more fun. Once you've practiced this technique, you can mentally estimate things that you can't calculate in your head. For example,

2389/27223 = ?

I think of this problem as 2.5 x 10³ divided by 2.7 x 10⁴

2.5/2.7 is slightly less than 1. 10³/10⁴ is exactly 10^-1.

So the quick estimate (no calculator) is a little below 1 x 10^-1, or 0.10

The calculated value was 0.088. Not bad for mental calculation, but the most important reason for this is to have a check on your calculations. Something significantly different than your estimate is likely to be warning you of a missed calculator step or entry.

You can impress, or worry, your friends when you use this approach to difficult calculations. It helps to have the metric prefixes in mind (e.g., milli-, mega-, tera-, etc.) and there associated 10^x meanings. I know that nano- is 10^-9 so if the result is 0.000000009 seconds, I can say 9 nanoseconds. A true nerd can incorporate the prefixes atto-, exa-, and fempto- into any party conversation, and be greeted with deep suspicion. But it is still fun.

I hope this helps,

Bob