Scientific notation offers us a way to express very small or very large, in the form of “times 10 to the power b”, where “a” is any number between 1 and 10 and “b” is an integer.
Here, we’ll be focusing first on the last part, the “10 to the b” part, or “10 raised to an integer”.
Now, let’s first look at “10 to the power 1, or first power. 10 to the power 1 means 10 times itself just 1 time, or simply 10.
Next lesson: Order of Operations with exponents
This transcript is only for your convenience. 10 squared means: 10 times itself 2 times, which is 100. 10 cubed means 10 times itself 3 times, which is 1000. And 10 to the power 4, or to the fourth power, means 10 times itself four times, which is 10000. Please note that as we increase our exponent, so increases our place value in the result. When we increase our exponent, it’s like we’re moving our decimal place a place to the right. Now let’s see what will happen when we’re raising 10 to negative exponents. To simplify this with a negative exponent, we’ll be taking the reciprocal, so we’ll be flipping it, so we’ll get the number that contains a positive exponent.
So, our reciprocal here would be 1 and this divided by 10 to power one, or simply 1/10. And 1/10 written as a decimal makes 0.1.
Let’s do the same thing for 10 to the power -2. So, we’ll take the reciprocal that has a positive exponent. 1 divided by 10 squared. This is 1 divided by 10 squared (which is 10 times 10), which is 100/ This is 0.01 written as a decimal. Then, 10 to the power -3 is 1 divided by 10 to the power 3, which is simply 1 divided by 1000, which is 0.001. And finally, 10 to the power -4. This is 1 divided by 10 to the fourth power, which makes 1 divided by 10000, which is 0.0001.
So here we looked at this since, in scientific notation, we can take any number, like 3, times 10 to the 2nd power.
Now, what 3 times 10 to the 2nd power is, it’s really three times 10 squared (and this is 100), and three times 100 is 300.
Now, what we also can do is, because it is “times 10 to the 2nd power”, we can take the decimal on 3, and move it two places to the right because this is also what happened with our 10s raised to different exponents. We just moved our decimal 1 place to the right when our exponent increased.
So, here we may take the decimal on 3 which, when you cannot find one, it is at the end of that number. Then we move it 2 places to the right and get our answer 300. So 3 times 10 to the 2nd power would result in 300.
Now let’s take a look at a number that’s raised to a negative exponent. For example, 4 times 10 to the power -3.
Here, when we did 10 to the power -1, then 10 to the -2, and so on, please note that as the exponents decreased, also our values decreased each time by one decimal place. Each time the numbers became smaller and smaller and smaller, each time the decimal moved one place to the left.
So, here we’ll be doing the same thing when we do “4 times 10 to the -3”. This would mean 4 times, and as we’ve seen, 10 to the -3 is 0.001. So this makes 4 times 1/1000, and this is 4 times 0.001, which makes 0.004.
And another way to deal with this is to just take that to the -3 power. This -3 power only means you’ll have to move the decimal 3 places to the left. Here, we would take the 4, and the decimal is behind it. Now just the decimal 3 places to the left. Now, this gives us the same result we had when we did it the other way.