Exponents and Multiplication – Product of Powers Property 04:47 minutes

Video Transcript

Transcript Exponents and Multiplication – Product of Powers Property

While Ethan chills on the couch, he chats with a friend. Oh no! Let’s take a look at the assignment and see what Ethan needs to do… He needs to write an essay for his English class. Ethan's English teacher, who's also a math enthusiast, gave the students 10² days to write 10³ words EACH DAY about courage. Ethan doesn't think it's so bad. Is he right?

Exponents and multiplication

Using exponents and multiplication, let’s help Ethan figure out how many words he needs to write per day. To make your life easier when multiplying exponents, you can use the Product of Powers rule. When you have a number raised to a power, as in aⁿ, multiplied by another number with the same base, aᵐ, you can write it in long form as 'a' multiplied 'n' times multiplied by 'a' multiplied 'm' times. All together, we can write this product as 'a' multiplied 'n' plus 'm' times.

Add and simplify

Or simply, aⁿ⁺ᵐ. To multiply exponents with the same base, add the exponents, then simplify. So… to calculate the number of words Ethan needs to write in two days, multiply the number of days the teacher assigned to write the essay times the number of words Ethan has to write. (10²)(10³) If you want, you can write out all the tens in long form as we showed you before.

The product of Powers rule

As you can see the 10 is multiplied 2 + 3 times, or simplified 5 times. We can write this as 10⁵.... or you can use the Product of Powers rule and just add the exponents since the base is the same for the numbers we want to multiply. Either way, the answer is 10⁵...or 100,000 words.That’s words, not characters.

Uh oh, that’s way more than Ethan expected. When he got the assignment, the word count seemed small. Feeling a bit dejected, he starts to write. After several hours, he realizes that he can write 1,024 or 2¹⁰ words per hour, and so far, he's written about 25,000 words. He still has 2⁴, or 16, hours to go. Will he get to 100,000 words in time? To solve this, calculate (2⁴)(2¹⁰). You can list the 2s in long form and then multiply, if it helps.

Same bases

As you can see, the 2 is multiplied 10 + 4, or simply, 14 times. You can write this as 2¹⁴...or since the bases are the same, you can use the Product of Powers rule again. (2¹⁰)(2⁴) = 2¹⁴...and that's equal to 16,384 words. We can also write this as 1.6384E4. E4? You might be wondering, what the heck is E4? It's shorthand for 10⁴. Now Ethan understands how to use the Product of Powers rule, but that doesn’t solve his problem...

Solve the problem

With the time he has left to write, he will have about 40,000 words altogether. Oh no! He'll be short 60,000 words, but Ethan has an idea – it’s a power move for sure…Will it work? Probably it won’t, but who knows? Maybe..., just maybe...