Metric Conversions
No Exponents Allowed in the Answer
Fifteen Examples

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The goal for the answer to every example in this section is this:

use appropriate metric prefixes to write the given measurement without use of exponents.

Another way to describe the goal is this:

to have the numerical part be not less than 1 and not greater than (or equal to) 1000.

In other words, you're not going to change to any particular metric prefix specified in the problem. You're going to a prefix that will cause the numerical portion to be greater than (or equal) to one and less than 1000. (You sometimes see it given as between 1 and 999.)

Please keep in mind that this goal is often not explicitly stated in the problem. You'll see that in the first example.

By the way, the rationale behind the 1-1000 thing is that metric exponents are separated from each other by some factor of 1000. For example, milli- differs from micro- by a factor of 1000 or fermto- differs from nano- by a factor of 10⁹ or 1000 million. (I am aware of metric prefixes that are not based on a factor of 100 (deci-, hecto-, etc. I'm ignoring them.)

The first ten examples are similar, in that the base unit is used (meters, liters, etc.) in the problem statement. By base unit, I mean one that does not have a prefix. Starting in example #11, I convert values that already have a prefix, we just need to go to another prefix so as to remove the exponent.

Example #1: Express 3.31 x 10¯⁹ m using a prefix.

Solution:

The correct answer to the above is 3.31 nm. Here's how (formatted two different ways, you need to be familiar with both):

(3.31 x 10¯⁹ m) (10⁹ nm / m) = 3.31 nm

		109 nm
3.31 x 10¯9 m	x	–––––––	= 3.31 nm
		1 m

If you selected micro- (the prefix before nano-), this is the answer:

(3.31 x 10¯⁹ m) (10⁶ μm / m) = 3.31 x 10¯³ μm = 0.00331 μm

Note that the numerical value is 1000 times smaller than the correct answer.

If you selected pico- (the prefix after nano-), you get this:

(3.31 x 10¯⁹ m) (10¹² pm / m) = 3.31 x 10³ pm = 3310 pm

Notice that the numerical value is 1000 times greater than the correct answer

Using micro- and pico- both satisfy the exact wording of the problem, but they both fail the unstated part of having the numerical part be between 1 and 999.

Example #2: Express 1 x 10⁷ volts using a prefix.

Solution:

Examining a set of metric prefixes, we see that mega- (10⁶) is the closest prefix, going in a "downward" direction. We need to convert our value in volts to the corresponding megavolt value:

(1 x 10⁷ V) (1 MV / 10⁶ V) = 10 MV

		1 MV
1 x 107 V	x	–––––––	= 10 MV
		106 V

Note that it is the closest prefix less than (or equal to) the exponent in the value in the problem.

Example #3: Express 4.8 x 10¯⁵ Joules using a prefix.

Solution:

Look at the 10¯⁵. We want the next prefix going "downward" in a list of metric prefixes.

That prefix is micro- (10¯⁶)

We convert from J to μJ:

(4.8 x 10¯⁵ J) (10⁶ μJ / 1 J) = 48 μJ

Note that this "downward" direction is in reference to a list of metric prefixes having the positive exponents listed first and the proceeding to the negative exponents. This link shows an example of what I mean.

Example #4a: Express 3.3 x 10¹¹ Hz using a prefix.

Solution:

From 10¹¹, the next prefix in a downward direction is giga- (10⁹)

(3.3 x 10¹¹ Hz) (1 GHz / 10⁹ Hz) = 330 GHz

		1 GHz
3.3 x 1011 Hz	x	–––––––	= 330 GHz
		109 Hz

Note that the unit of tera- (10¹²) is closer to the exponent in 10¹¹, but we want the next unit smaller than our exponent. If we had converted to tera-, this is what would have resulted:

(3.3 x 10¹¹ Hz) (1 THz / 10¹² Hz) = 0.33 THz

While there is absolutely nothing wrong with 0.33 THz, the 0.33 value fails the (sometimes unstated) stipulation that the numerical value must be greater than (or equal to) 1.

Example #4b: Express 3.3 x 10¹¹ Hz using a prefix.

Solution:

Another way to do the change is to move the exponent (in this case, the 10¹¹) to the nearest exponent that will keep the numerical part between 1 and less than 1000.

What that means is that I will divide the exponent and multiply the numerical part. With 10¹¹, I would divide the exponent by 10² and multiply the numerical part by 10²:

3.3 x 10¹¹ Hz = [3.3 x 10²] x [10¹¹ / 10²] Hz = 330 x 10⁹ Hz

Then, I get rid of the prefix associated with 10⁹:

		1 GHz
330 x 109 Hz	x	–––––––	= 330 GHz
		109 Hz

Example #5: Express 5500 Ångstroms (symbol is Å) using a prefix.

Solution #1:

This is a bit of an odd one because Å is not a SI-unit. However, we can express Å using a SI-unit:

(5500 Å) (10¯¹⁰ m / 1 Å) = 5.5 x 10¯⁷ m

Looking at the exponent in 10¯⁷, we select nano- (10¯⁹) as the next unit in the downward (or smaller) direction:

(5.5 x 10¯⁷ m) (10⁹ nm / 1 m) = 550 nm

Solution #2:

Ångstrom's original definition of what one Å equalled was this:

1 Å = 10¯⁸ cm

Suppose you were given that conversion in the body of the problem. What to do?

You could convert 10¯⁸ cm / Å to 10¯¹⁰ m / Å and continue with solution #1. Suppose you didn't. What to do?

Let's convert and see what happens:

(5500 Å) (10¯⁸ cm / Å) = 5.5 x 10¯⁵ cm

We have an exponent (but we do have a numerical value between 1 and 1000). Since we need the exponent gone, we have to keep converting. Hopefully, you can see that moving to the next smaller prefix (milli-) isn't sufficient. Perhaps, going to micro- would work:

(5.5 x 10¯⁵ cm) (10⁴ μm / 1 cm) = 0.55 μm

Well, that didn't work. So, we try nano-:

(5.5 x 10¯⁵ cm) (10⁷ nm / 1 cm) = 550 nm

Comment: notice how the exponents are different. In the micro- conversion, I used 10⁴ rather than 10⁶. And, in the nano- conversion I used 10⁷ rather than 10⁹.

Example #6: Use the appropriate metric prefixes to write the following measurements without the use of exponents:

6.5 x 10¯⁶ m

Solution:

1) The next prefix "down" from 10¯⁶ is nano-. This means we will convert our answer to nanometers.

		109 nm
6.5 x 10¯6 m	x	–––––––	= 6500 nm
		1 m

2) Oops. A bit of an overshoot. Back up to the micro- prefix using 6500 nm.

		1 μm
6500 nm	x	–––––––	= 6.5 μm
		103 nm

Note that we wound up using the prefix that is associated with the exponent.

3) If we started from the value given in the problem, we'd have this:

		106 μm
6.5 x 10¯6 m	x	–––––––	= 6.5 μm
		1 m

4) When the exponent of the value to be converted is associated with a unit, you should use that exponent. For example, convert 6.5 x 10¯¹² m.

		1012 pm
6.5 x 10¯12 m	x	–––––––	= 6.5 pm
		1 m

Example #7: Use a prefix that eliminates the exponent from 6.5 x 10⁶ m.

1) The exponent of 6 is associated with the prefix mega-. Let's use it in a conversion and see what happens.

		1 Mm
6.5 x 106 m	x	–––––––	= 6.5 Mm
		106 m

2) If the exponent in the problem is associated with a prefix, then use that prefix. Don't move down to the next prefix. If you do, the answer will be 1000 or greater.

3) Here is this problem done with the next prefix down, which is kilo-. Remember, choosing kilo- is an incorrect choice.

		1 km
6.5 x 106 m	x	–––––––	= 6500 km (or 6.5 x 103 km)
		103 m

4) Some other examples of a prefix being associated with an exponent are:

(a) centi- is associated with 10¯²
(b) nano- is associated with 10¯⁹
(c) tera- is associated with 10¹²

Example #8: 6.35 x 10¯⁴ L

Solution:

1) I know I have to go to the next prefix down the list (remember, the list has to run from most positive to most negative). Let's change the exponent to to the next most negative exponent from the list:

6.35 x 10¯⁴ L = 635 x 10¯⁶ L

I changed the exponent from −4 to −6, which is the same as a division by 100. To counteract that division, I multiplied the 6.35 by 100 to give 635.

This modification gets me a number (the 635) greater than 1 and less than 1000.

2) The prefix associated with 10¯⁶ is micro-:

		106 μL
635 x 10¯6 L	x	–––––––	= 635 μL
		1 L

3) Suppose you changed the exponent from −4 to −3 (remember, this is an incorrect choice):

6.35 x 10¯⁴ L = [6.35 / 10] x [10¯⁴ x 10] L = 0.635 x 10¯³ L

If I now get rid of the exponent of 10¯³, I wind up with an answer of 0.635 mL.

This is not what I wanted. I agree that it is being written with a prefix. It's just not within the (often unstated) limits of larger than 1 and less than 1000.

Example #9: 3.5 x 10¯¹¹ s

Solution:

1) The next most negative metric prefix going in a downward direction on a list would be pico- (10¯¹²). Let's convert to it:

[3.5 x 10] x [10¯¹¹ / 10] s = 35 x 10¯¹² s

2) Get rid of the exponent:

		1012 ps
35 x 10¯12 s	x	–––––––	= 35 ps
		1 s

Example #10: 3.5 x 10¯¹⁰ g

Solution:

1) I'm going to deliberately change to the "wrong" exponent:

[3.5 / 10] x [10¯¹⁰ x 10] g = 0.35 x 10¯⁹ g

2) let's say we don't recognize the mistake yet, so we continue by getting rid of the exponent with the appropriate prefix (nano-):

		109 ng
0.35 x 10¯9 g	x	–––––––	= 0.35 ng
		1 g

3) I now realize my mistake. That 0.35 has to be changed. The next prefix that is more negative is pico- (10¯¹²):

		103 pg
0.35 ng	x	–––––––	= 350 pg
		1 ng

4) The conversion from the original 10¯¹⁰ to 10¯¹² would be this:

[3.5 x 100] x [10¯¹⁰ / 100] g = 350 x 10¯¹² g

You can finish it from there, if you so desire.

Example #11: Express 6.54 x 10⁹ fs using a prefix (and no exponent).

Solution:

1) See that exponent of 9? Knowing that femto- is associated with 10¯¹⁵, is there a prefix that is an "exponential distance" of 9 away from femto-?

2) Why yes, yes there is. It's the unit micro- (10¯⁶). Let's see what happens:

		1 μs
6.54 x 109 fs	x	–––––––	= 6.54 μs
		109 fs

3) Another technique is to convert to the base unit and then eliminiate the exponent. Here's the sequence written out without any supporting work:

6.54 x 10⁹ fs = 6.54 x 10¯⁶ s = 6.54 μs

4) Here is the conversion written in a dimensional analysis style:

		1 s		106 μs
6.54 x 109 fs	x	–––––––	x	–––––––	= 6.54 μs
		1015 fs		1 s

Notice how there is a total exponent of 10¹⁵ in the numerator and 10¹⁵ in the denominator.

5) Comment: the original problem that was in my notes did not include the phrase 'and no exponent.' The original problem wording is interesting in that this:

Express 6.54 x 10⁹ fs using a prefix

can be answered correctly in this manner:

6.54 x 10⁹ fs

or this:

6.54 x 10⁶ ps

or this:

6.54 x 10³ ns

The real goal of the question (not written by the ChemTeam, BTW) is to eliminate the exponent, not write the answer with just any old prefix you want.

Example #12: Convert 12.5 x 10¯⁸ kg to a unit that does not use an exponent.

Solution:

1) Convert to the base unit:

		103 g
12.5 x 10¯8 kg	x	–––––––	= 12.5 x 10¯5 g
		1 kg

2) There is a problem. We need an exponent associated with a metric prefix and 10¯⁵ is not suitable. However, 10¯⁶ is.

[12.5 x 10] x [10¯⁵ / 10] g = 125 x 10¯⁶ g

3) Now, we can eliminate the exponent:

		106 μg
125 x 10¯6 g	x	–––––––	= 125 μg
		1 g

Example #13: Eliminate the exponent for the following value by using a prefix: 1.16 10¯⁷ kg.

Solution:

1) Take it to the base unit:

		103 g
1.16 x 10¯7 kg	x	–––––––	= 1.16 x 10¯4 g
		1 kg

2) Adjust the exponent to correspond with the metric prefix value next "downward" from 10¯⁴:

1.16 x 10¯⁴ g = [1.16 x 100] x [10¯⁴ / 100] g = 116 x 10¯⁶ g = 116 μg

3) We know that 10¯⁶ is associated with micro-. Here's the work to support the answer of 116 μg:

		106 μg
116 x 10¯6 g	x	–––––––	= 116 μg
		1 g

Example #14: 3.4 x 10¯¹⁰ Tg

Solution:

1) Let's take the exponent from 10¯¹⁰ down to 10¯¹²:

[3.4 x 100] x [10¯¹⁰/100] Tg = 340 x 10¯¹² Tg

2) Tera- is 10¹² and we have 10¯¹² in the exponent. Thinking 340 g is the answer? You'd be correct:

		1012 g
340 x 10¯12 Tg	x	–––––––	= 340 g
		1 Tg

Example #15: Eliminate the exponent of 6.44 x 10⁹ Ms by using a prefix.

Solution:

1) To the base unit, my lovelies!

		106 s
6.44 x 109 Ms	x	–––––––	= 6.44 x 1015 s
		1 Ms

2) The metric prefix associated with 10¹⁵ is peta-. Convert to it:

		1 Ps
6.44 x 1015 s	x	–––––––	= 6.44 Ps
		1015 s

3) You may want to try setting up a direct conversion yourself. By the way, 1 Ps equals 3.171 x 10⁷ years.

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