When we multiply by powers of ten, it means we are multiplying by numbers like $10,100,1000,\dots$10,100,1000,…. The number in our power tells us how many factors of ten we need to multiply by. So, for $3\times10^4$3×104 we are multiplying $3$3 by $10$10, four times. This means we are solving $3\times10\times10\times10\times10$3×10×10×10×10
The first video reminds you of how this helps us when multiplying numbers, and the important role of place value. It also shows you a simple example of multiplying a decimal by a power of $10$10.
In the second video, we are multiplying by numbers with more powers of $10$10, as well as looking at decimal numbers with more than one digit. We use the same process, and placeholders for zero become really important here. Without a $0$0 in some of our place columns, our answer might be very different.
In the last video, we start with an example of multiplying a number by up to $10^6$106, so you can see we follow the same process. We just move more times to the left. We also look at how each digit stays in the same order, so once you move the first digit, you can see the others follow right behind.
This is the same as multiplying whole numbers by powers of ten, we just have some different place value columns.
Write $10$10 as an appropriate power of 10.
What is $0.5\times10^2$0.5×102?