Common Core Standards: Math

Expressions and Equations 8.EE.A.4

4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Students should be aware that we can write really big or really small numbers as single digits times a power of 10. We call this scientific notation.

The basic idea is that a number containing a lot of zeros at the end can be rewritten as the product of a number between one and ten and a power of 10. It's so much easier, for example, to write 5 × 1011 than to write 500,000,000,000. And when those numbers are used in computation, using scientific notation is an excellent way to save your time and effort.

Students should already know how to convert numbers into this format. They should also understand that they can't multiply any number by a power of 10 and have it be scientific notation. The number has to be between 1 and 10. Be sure they can also convert numbers like 24,900,000 and 0.0000617 into proper scientific notation. (Those are 2.49 × 107 and 6.17 × 10-5 in case you were wondering).

It's also important that students make use of scientific notation and understand appropriate units. For instance, a distance of 3.1 × 107 millimeters would probably best be expressed in kilometers (that makes sense considering that 1 × 106 mm = 1 km).

Drills

1. What is six trillion in scientific notation?

6 × 1012

Six trillion is 6,000,000,000,000 or 6 followed by twelve zeros. The decimal point moves 12 places. That means our 10 should have a power of 12. All the other answers have the wrong number of zeros.

2. What is 2.3 billion in scientific notation?

2.3 × 109

First, we can expand 2.3 billion so that it's 2,300,000,000 or 2.3 followed by eight zeros. To move the decimal in between the 2 and the 3, we have to move it nine places, meaning the power of 10 is 9. That gives us an answer of 2.3 × 109.

3. Which number is represented by 5.61 × 104?

56,100

To multiply by 10 to the fourth power, move the decimal four places to the right. Our final answer should have an additional two zeros at the end of the given numbers, which means (C) is what we're looking for.

4. The population of Los Angeles is about 3.83 million. What is this number in scientific notation?

3.83 × 106

The key to scientific notation is to move the decimal point so that we make a number in between 1 and 10. So we want to move the decimal point in 3,830,000 between the first 3 and the 8. To do that, we have to move it six decimal places to make 3.83 × 106. While (D) is technically the same value, it is not in proper scientific notation because 383 isn't between 1 and 10.

5. Uranium-238 has a half-life of 4.5 billion years. What is this half-life in scientific notation?

4.5 × 109

First of all, the billion in question is 4.5, not 238. The expanded number is 4,500,000,000 and we want to move the decimal point over 9 places. This means we should have 4.5 × 109 as our answer.

6. Express (4 × 105)(9 × 103) in scientific notation.

3.6 × 109

The first thing we want to do is multiply 4 × 9 = 36. Now, we know that 105 × 103 = 108. Even though 36 × 108 is correct, the 36 is too big so we have to convert 36 × 108 to 3.6 × 10 × 108 and then to 3.6 × 109.

7. Express (4 × 105)2 in scientific notation.

1.6 × 1011

Don't let the outer exponent confuse you! This is the same as (4 × 105)(4 × 105). We know 4 × 4 = 16 and 105 × 105 = 1010, so we end up with 16 × 1010, but of course, we need to make sure to change it to 1.6 × 1011.

8. Before he decided to cut down on the carbs, an elephant weighed 27,000 pounds. What is this number in scientific notation?

2.7 × 104 pounds

If you start with 2.7, and move the decimal four places to the right, you'll get 27,000. In other words, we should get 2.7 × 104. Make sure your units are right, too!

9. The weight of NASA's Hubble Telescope is approximately 24,500 pounds. What is this weight in scientific notation?

2.45 × 104 pounds

From 2.45, the decimal needs to be moved four places to the right to make 24,500. We can automatically see that (B) and (D) aren't in proper scientific notation even though they're the correct value, and (A) would only translate to 245, not 24,500.

10. The depth of the Mariana Trench in the Pacific Ocean is approximately 35,840 feet. What is this number in scientific notation?

3.584 × 104 feet