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Exponent Review

Scientific Notation

This week your student will learn the rules for multiplying and dividing expressions with exponents. Exponents are a way of keeping track of how many times a number has been repeatedly multiplied. For example, instead of writing 8⋅8⋅8⋅8⋅8⋅8⋅8, we can write 87 instead. The number repeatedly multiplied is called the base, which in this example is 8. The 7 here is called the exponent.

Using our understanding of repeated multiplication, we’ll figure out several “rules” for exponents. For example, suppose we want to understand the expression 103⋅104. Rewriting this to show all the factors, we get (10⋅10⋅10)⋅(10⋅10⋅10⋅10). Since this is really 7 10s multiplied together, we can write 103⋅104=107. By counting the repeated factors that are 10, we’ve added the exponents together (there are 3 of them, and then 4 more). This leads us to understanding a more general rule about exponents; when multiplying powers of the same base, we add the exponents together:

Using similar reasoning, we can figure out that when working with powers of powers, we multiply the exponents together:

These patterns will lead to other discoveries later on.

Here is a task to try with your student:

Solution:

This week your student will use powers of 10 to work with very large or very small numbers. For example, the United States mint has made over 500,000,000,000 pennies. In order to understand this number, we have to count all of the zeros. Since there are 11 of them this means there are 500 billion pennies. Using powers of 10, we can write this as 5⋅1011. The advantage to this way of writing the number is that we can see right away how many zeros there are (11), and more efficiently compare numbers when they are both written in this form. The same is true for small quantities. For example, a single atom of carbon weighs about 0.0000000000000000000000199 grams. If we write this using powers of 10, it becomes (1.99)⋅10−23.

Not only do powers of 10 make it easier to write this number, but they also help avoid errors since it would be very easy to add or take away a zero when writing out the decimal without realizing! Writing numbers in this way is called scientific notation. We can use the exponent rules learned earlier to estimate and solve problems with scientific notation.

This table shows the top speeds of different vehicles.

Solutions

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.