Properties of Exponents

The lesson below is still images taken from the fully animated PowerPoint I created that is available in my Teachers Pay Teachers store page. If the PowerPoint is not one of my freebies, you may also head over to my YouTube channel to see the slideshow fully animated and can pause as needed to be sure you grasp each concept before moving forward (each lesson will always be free there!).

In this lesson we explore the different properties exponents have when multiplying or dividing with the same base. The slides break down each operation into a visual depiction of what is going on to reduce memorization while increasing understanding. Students can see exponents multiplied, divided, and raised to another power. Then we dive into negative exponents and how they are useful.

So without further ado, read through the slides below to get a feel for how use properties of exponents!

Prop 1Prop 2Prop 3Prop 4Prop 5Prop 6Prop 7Prop 8Prop 9

Phew, that’s a lot to take in. Once you’ve gone over this and found some practice problems to cement the idea in your head what integers are, you may move on to the next lesson below!

Scientific Notation

Just in case the slides aren’t your thing, here is a text outline of the main points of the lessons above!

  • Exponents
  • Properties
  • Objectives
  • By the end of this lesson you should feel comfortable:
  • Multiplying with exponents
  • Dividing with exponents
  • Raising exponents to powers
  • Utilizing negative exponents
  • Multiplying with Exponents
  • When working with exponents, just like with many things in math, having like terms is important.
  • In this case, we are looking for the same base, the number being raised to a power.
  • So what does this mean?
  • First, 3 raised to the 2nd power is:
  • Then 3 raised to the 3rd power is:
  • But five 3’s is the same as:
  • Multiplying with Exponents
  • Hmm as always, this seems long and annoying if we are working with very large numbers.
  • Perhaps there is an easier way?
  • We had two 3’s and three 3’s, which gave us a total of five 3’s.
  • Total means to add, though, right?
  • And sure enough, all we have to do is add our exponents when multiplying with the same base!
  • Dividing with exponents
  • So you’ve probably guessed from the title that if we can multiply with exponents, we can divide as well.
  • Sure enough:
  • Again, let’s observe what’s really represented here.
  • 6 to the 4th power is:
  • And 6 to the 2nd power is:
  • But things on top and bottom cancel because anything over itself is just 1.
  • So all that’s left is two 6’s on top, which is the same as:
  • Dividing with exponents
  • As always, we should probably find a quicker way to handle these. We are, after all, lazy mathematicians.
  • We ended up with only two left over. And left over means that we subtracted.
  • Sure enough, if we subtract our exponents
  • 4-2=2
  • Tada! You can now divide with exponents.
  • Raising Exponents to Exponents
  • But there is still yet another way to operate with exponents as the title suggests:
  • So something, four squared, is being raised to the third power, or:
  • Now we can see that there are six 4’s, which is the same as:
  • Buuuuuuut, can we make things easier?
  • If we have groups, that means we multiply!
  • So 2 x 3 = 6
  • Negative Exponents
  • Lastly, we need to discuss a different type of exponent, negative numbers!
  • We can have a negative number in a situation such as this:
  • We know from experience, that we can subtract 2-5= -3, but what does this mean?
  • So we have 3 6’s left on the bottom, or:
  • But as we said, 2-5 = -3, so we also know that 1 over 6 to the third power can be written as:
  • Negative Exponents
  • Ok, so that’s a fair bit to take in, so let’s try to break it down into simpler terms.
  • The easiest way to remember the rule for negative exponents is that they flip to the bottom to become positive.
  • Here we see that 6 raised to the negative 3rd power is the same as 1 over 6 raised to the positive 3rd power.
  • As always, negatives are opposites, which in this case means:
  • Flip it to the bottom!
  • conclusion
  • When multiplying two numbers with exponents that have the same base, you add the exponents.
  • When dividing two numbers with exponents that have the same base, you subtract the exponents
  • When raising a number with an exponent to a power, you multiply the exponents.
  • A negative exponent can be rewritten as a positive exponent under 1.
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