4.5 SAT Math - Scientific Notation

There's not much scientific notation on the SAT math section, but you should know how to use it.  Basically, scientific notation takes really big or really small numbers and rights them as a number times ten to a certain power.  So instead of writing  4,323,000,000,000 you would write 4.323 x 10¹².  Instead of .00003102  you would write 3.102 x 10^-5.  So a small number gets a negative exponent; a big number gets a positive exponent.

Scientific notation always consists of a number in the ones place, followed by the necessary decimal to cover the other places once it's multiplied back out.  The exponent that goes with the ten is determined by the number of spaces you needed to move the decimal from its original spot to where it ends up in scientific notation.

-SAT Math Tip: The most common and trickiest testing of scientific notation comes from requiring you to fix your answer. 

Example Problem

5.  Mercury is the smallest planet in terms of volume, but extremely heavy, with an approximate mass of 3.302 x 1023km.  If Jupiter, the largest planet, has an approximate mass of 1.8987 x 1027 km, what is the approximate ratio of the largest planet to the smallest?

(A) 1.74 x 10+4
(B) 8.58 x 100
(C) 5.75 x 103
(D) 1.74 x 104
(E) .575 x 105

To solve, set it up like any other ratio:

, divide the "regular numbers" by each other to get | .575 , then work with the exponents according to exponent rules, which means when regular numbers are divided, exponents are subtracted.  So you're left with 10⁴.  Final answer? .575 x 10⁴.  Not too difficult. 

What's the problem?  Why isn't that answer an option?  Because you're not done.  Proper scientific notation has one number in the ones place, and then decimals for the rest.  You need to move that decimal over one place to make your "regular number" 5.75.  That's also going to change your exponent.  Remember: a small number being turned into scientific notation (a decimal less than one) gets a negative exponent; that means, we've moved the decimal -1 places, or subtracting one from our original exponent.  So the real final answer?  5.75 x 10³.

-SAT Math Hint: If you get small enough numbers (like something times 10³) type it into your calculator to double-check.  Your calculator will give you something like 4.23 ∈ 12 when a number is too big, which means it's giving you scientific notation (4.23 ∈ 12 means 4.23 x 10¹²) but if it's a small enough number, your calculator will convert it back to standard notation if you type it in scientific.

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