Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
Steps to Solve
Let's look at how to multiply 100 x 1000. The process to solve this problem is actually quite simple; to multiply 100 times 1000, take the following steps:
- Count the number of zeros in 100.
- Count the number of zeros in 1000.
- Add up the number of zeros you counted in step 1 and 2.
- Write the number 1, and then follow it with the number of zeros you found in step 3.
Notice that each step is very easy to perform. That's the great thing about breaking a problem down into steps - if you just take it one step at a time, it's really quite simple!
Let's take the problem through the steps to find our answer. The first step is to count the number of zeros in 100 - there are 2 zeros in 100. The next step is to count the number of zeros in 1000 - there are 3 zeros in 1000. Now, add up the number of zeros we found in steps 1 and 2:
2 + 3 = 5
We have 5 zeros total. The last step is to write the number 1 and follow it with the number of zeros found, or 5 zeros:
This is our answer!
When we multiply 100 by 1000, we get 100,000.
Solving Method Explained Using Exponents.
You're probably glad to know how to solve the problem, but you may be wondering why this process works. Well, don't worry! We're going to explain it using exponents.
To understand this process, we need to be familiar with a couple of facts. The first is writing 100 and 1000 as powers of 10. Observe the following:
100 = 10 x 10 = 10 2
1000 = 10 x 10 x 10 = 10 3
We know that 100 = 10 2 and that 1000 = 10 3. Notice that the exponents are equal to the number of zeros in the number. Basically, to write these numbers as powers of 10, count the number of zeros in the number, then raise 10 to that power. This corresponds to the first and second steps of the solving process, and we find the following:
100 x 1000 = 10 2 x 10 3
The next fact that we need to be familiar with is the multiplication rule of exponents. This rule states that
We know that a b x a c = a b+c. We can apply this to multiplying 10 2 x 10 3. This corresponds to the third step of the solving process, and we find the following:
100 x 1000 = 10 2 x 10 3 = 10 2+3 = 10 5
The last step in the solving process is calculating 10 5, which is 100000. Altogether, we find that:
100 x 1000 = 10 2 x 10 3 = 10 2+3 = 10 5 = 100000
Pretty neat, huh? We basically just walked through an informal proof that 100 x 1000 = 100000. What's even better is that this process of solving can be extended to any multiplication problem that multiplies powers of 10 together.
For example, suppose you enter a contest to win ten thousand $100 bills. You want to know how much money that actually is, so you need to multiply 100 x 10000. These are both powers of 10, so you can take it through the steps outlined in the first section of this lesson to solve.
First, count the number of zeros in 100, which is 2. Then, count the number of zeros in 10000, which is 4. Add up those zeros to get 2 + 4 = 6. Finally, write the number 1 and follow it with 6 zeros to get 1000000. We see that 100 x 10000 = 1000000. Breaking this down into powers of ten and using the exponent rule shows us the following:
100 x 10000 = 10 2 x 10 4 = 10 2+4 = 10 6 = 1000000
We see we get the same answer, which was to be expected. This tells you that if you win the contest, you get $1,000,000! Wow! Here's hoping!
Multiplying 100 x 1000 is really pretty easy when we break it down into steps, as is any multiplication problem involving powers of 10. This is a great process to put to memory for future reference!
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