SAT Tip of the Week: Scientific Notation and Exponents

SAT Tip of the Week - FullScientific Notation is not a heavily-tested concept on the SAT (whew!) – and you might not see it at all! But it’s definitely useful to review, and it’s a concept that goes hand-in-hand with understanding exponents.

Large numbers and very small decimals are often expressed with exponents using scientific notation. Scientific notation means expressing a number as a product of a decimal and the number 10 raised to a certain power. The reason scientific notation is used is that is saves space. Who would want to write .000000000000000000547, when 5.47 x 1019 saves us a lot more room?

The value of the exponent indicates the number of places the decimal moves. In our example above, we moved the decimal 19 places to the right, so the exponent was a positive 19.

108 = 1 + 8 zeros = 100,000,000

.076 x 105 = 7600 (the decimal moves five places to the right)

.0000000577 x 106 = .0577 (the decimal moves six places to the right)

9.6 x 10-4 = .00096

Here’s the “secret key” to solving any scientific notation question: positive exponents move decimals to the right; negative exponents move to the decimal to the left. Let’s look at a practice SAT question:

A pitcher throws a ball in 4.4 x 10-7 seconds. A second pitcher throws a ball 100 times slower than the first pitcher. How long did the second pitcher’s throw take, in seconds?

(A) 4.4 x 10-107

(B) 4.4 x 10-14

(C) 4.4 x 10-9

(D) 4.4 x 10-5

(E) 4.4 x 10-7/2

Let’s start by writing out 4.4 x 10-7. Since we have a negative exponent, we know the decimal will move to the left. .00000044 = 4.4 x 10-7. Now we would multiply the decimal by 100. Since there are two zeroes in 100, the decimal will move two places to the right. The answer would be .000044. To rewrite that in Scientific Notation, we can move the decimal 5 places to the right again, which would be a negative exponent of 5. The answer is (D).

A faster way to think about this question is to know that 100 = 102. We can express the solution as: 4.4 x 10-7 x 102. If you remember your exponent rules, when we multiply exponents with the same base, we can add the exponents. -7 + 2 = -5. Again, this matches choice (D).

Remember if you have to divide two numbers written in scientific notation, you can subtract the exponents (know your exponent rules!). For example, 4.4 x 10-7 divided by 2.2 x 10-9 = (4.4/2.2) x 10(-7)-(-9) = 2 x 10(-7)+9 = 2 x 102 = 200. Likewise, if you’re multiplying in scientific notation, you can add the exponents (the answer in the image above would be 9). Much easier than writing out all those zeros!

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Vivian Kerr is a regular contributor to the Veritas Prep blog, providing advice to help students better prepare for the GMAT and the SAT.