Powers of 10 and Scientific Notation

Many GMAT and Executive Assessment (EA) exponents problems involve the use of 10 as a base. Since all the math we do is in a base 10 system, multiplying or dividing by powers of 10 simply moves the decimal point of a number. This enables us to notate very large and very small numbers with multiplication by powers of 10, a method called “scientific notation.” Problems featuring scientific notation are quite common on the GMAT and EA, although the tests do not use the term.

Let’s say you want to notate the 2019 GDP of the United States without using the word “trillion.” The value in question is 21.43 trillion dollars. A trillion is a million million and requires 12 zeros! (though here there are only 10 because of the .43)

21.43 trillion = 21,430,000,000,000 

These zeros are getting a bit out of hand. Using scientific notation, we can represent the value this way:

2.143 * 1013

Conventions dictate that the coefficient be between 1 and 10, so we use 2.143 instead of 21.43 or 214.3. GMAT and EA questions will stick to this rule; all of the answer choices will be “proper” and correct or incorrect based on value alone, not style. Every multiplication by 10 effectively moves the decimal one place to the right, taking a new 0 on the number when necessary. It will take 13 of these moves to get back to the version of the “original” version of the number, so the exponent on the 10 in our notation is 13.

What if you want to tell someone how long light takes to travel 1 kilometer? The value is 0.0000033 seconds. The zeros are impressive, but we could simply write the number like this:

3.3 * 10-6

Since 10-1 = 1/10, multiplying by negative powers of 10 is the same thing as dividing by 10, which shifts the decimal point one place to the left. As before, zeros must be added to the number when we need them. In scientific notation, the decimal is 6 places to the right of where it would be in “normal” notation, so we need an exponent of -6 on the 10 to move the decimal back six places to the left.

Let’s try out some scientific notation problems:

A certain state’s milk production was 980 million pounds in 2077 and 2.7 billion pounds in 2014. Approximately how many million gallons of milk did the state produce in 2014 than in 2007? (1 billion = 109 and 1 gallon = 8.6 pounds.)

  1. 100
  2. 200
  3. 1,700
  4. 8,200
  5. 14,800

A certain state must have bought a lot of cows between 2007 and 2014. As we’ll see in the rest of our problems, many scientific notation problems on the GMAT and EA involve unit conversions – in this case from pounds to gallons. You’ll want to be familiar with the number of zeros indicated by million, billion, and trillion.

2.7 billion = 2.7 * 109

980 million = 9.8 * 108

The easiest way to handle operations with scientific notation is to work in columns. Deal with your coefficients and your powers of 10 separately, and convert back to “normal” notation as a final step when necessary. Since we are subtracting here (as indicated by the “how many more” phrase in the question), it makes sense to alter our scientific notation so that we are working with the same power of 10 in each number.

(2.7 * 109) – (9.8 * 108)

(27 * 108) – (9.8 * 108)

Technically we’ve “broken the rules” by using an exponent greater than 10 here, but that doesn’t matter. Now we can factor out the 108.

(27 * 108) – (9.8 * 108)

(27 – 9.8) * 108

17.2 * 108

This is still the difference in pounds, not gallons. To convert from pounds to gallons, we need to divide by 8.6. You may notice that 8.6 is exactly half of the 17.2 we’ve calculated. This is never coincidental and always a sign that you are on the right track.

(17.2 * 108) / 8.6

2 * 108

At this point, B must be the correct answer. Remember that the question asked us how many more million gallons were produced in 2014 than in 2007. That’s why the correct answer is 200, not 2 * 108.

2 * 108 = 200,000,000 = 200 million

Let’s try another:

A computer can perform 1,000,000 calculations per second. At this rate, how many hours will it take this computer to perform the 3.6 x 1011 calculations required to solve a certain problem?

  1. 60
  2. 100
  3. 600
  4. 1,000
  5. 6,000

Again, we have unit conversions. This time it is seconds to hours. GMAT and EA questions typically italicize units when conversions are involved to draw your attention to the shift.

If you know the seconds-to-hours conversion, this problem can be quite easy. There are 60 seconds in a minute and 60 minutes in an hour, so the number of seconds in an hour is 602 = 3600. This means that the computer in question performs 3600 * 1,000,000 calculations per hour.

(3.6 * 103) * (1 * 106) = 3.6 * 109

Switching to scientific notation and using your exponent rules to figure out the exponent on the 10 is generally better than writing out lots of zeros and hoping you count correctly. Observe that the number of calculations required to solve the “certain problem” we are asked about just happens to be 3.6 * 1011. This value is 100 times our calculated rate of 3.6 * 109 calculations per hour, so the correct answer is B, 100.

Let’s do just one more:

The age of certain granite rocks found in northwestern Canada is approximately 1.2 x 1017 seconds. Which of the following is closest to the age of these rocks, in years? (1 year is approximately 3.2 x 107 seconds.)

  1. 3.8 x 109
  2. 5.9 x 109
  3. 2.0 x 1010
  4. 2.0 x 1011
  5. 3.8 x  1011

Only in a GMAT/EA problem would anyone know the age in seconds of certain granite rocks found in northwestern Canada. Thankfully, this problem spares you the trouble of calculating the number of seconds in a year by supplying you with the value in scientific notation: 3.2 * 107.

The best way to perform division (or multiplication) on scientific notation values is to work “in columns,” taking the coefficients and the powers of 10 separately. Notice that your answer choices here stay in scientific notation, so converting to “normal” notation would just force you to switch back again at the end.

1.2 * 1017 / 3.2 * 107

1017 / 107 = 1010, but dividing 1.2 by 3.2 is somewhat less straightforward. If we recognize 0.4 as the greatest common factor of 1.2 and 3.2, we can see 1.2 / 3.2 as ⅜ or its decimal form: 0.375.

0.375 * 1010

Answers A and E both look good; they have simply moved the decimal one place to the right and rounded 3.75 to 3.8. But should the exponent be 9 (answer A) or 11 (answer E)? Since these answers involve moving the decimal one place to the right, it is now one place closer to where it would be in “normal” notation, and the correct change is shifting the exponent down to 9, not up to 11. The correct answer is A.

This concludes our foray into scientific notation. Next time we’ll see how powers of 10 can be used to notate numbers that are almost integers

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Contributor: Elijah Mize (Apex GMAT Instructor)

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