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Studying for a test? Prepare with these 2 lessons on Module 1: Integer exponents and scientific notation.
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Let's see if we can simplify this expression right over here and write it in scientific notation. So the first thing I want to do, well I already have parts of this expression that are written in scientific notation. For my brain, to simplify the multiplication, I like to write everything into scientific notation and then do whatever I need to do to get the final product into scientific notation. So this part right over here, 0.2, is not in scientific notation. In order for it to be scientific notation, it would have to be some number between 1 and 10, not including 10-- so greater than or equal to 1, less than 10-- being multiplied by some power of 10. And this is clearly less than 1. But we could just view this as, look, this is in the tenths place. This is 2 times 1/10. 1/10 is 10 to the negative 1. So this is the same thing as 2 times 10 to the negative 1-- same thing as 2 times 1/10. Now if we look in the denominator, in blue, we have this part. It is written in scientific notation. But this green part is not. But we could easily write it as-- this is five 10,000's. 10,000 is 10 to the fourth. So this is the same thing as 5 times 10 to the fourth power. And you see that it has one, two, three, four 0's. So now let's take the product in the numerator and the denominator. So in the numerator, I'm just going to swap the order in which I'm multiplying. I'm just multiplying a bunch of stuff. 4.6 times 10 to the sixth times 2 times 10 to the negative 1. It doesn't matter what order I multiply them in. So I could rewrite this as 4.6 times 2 times 2 times 10 to the sixth-- I'm switching colors-- times 10 to the negative 1. And then in the denominator, let me just write the 5 times 2.3 times 10 to the fourth times 10 to the negative 2. And now let us attempt to simplify this thing. So here, we have 4.6 times 2. Let me circle that. So 4.6 times 2 is 9.2. So that's 9.2. And then 10 to the sixth times 10 to the negative 1-- we have the same base. We're taking the product. We can add the exponents-- is going to be 10 to the 6 minus 1 or 10 to the fifth power. So we've simplified our numerator. And now in our denominator, let's see. 5 times 2.3-- 5 times 2 is 10. 5 times 0.3 is 1.5, so it's going to be 11.5. So this is going to be 11.5. And then if I multiply 10 to the fourth times 10 to the negative 2, that's going to be 10 to the 4 minus 2 or 10 squared-- times 10 to the second power. And now I can divide these two things. So this is going to be equal to-- we'll have to think about what 9.2 over 11.5 is. But actually let me just do that right now, get a little practice dividing decimals. Let me get some real estate here. Let me do that in the same color. 9.2 divided by 11.5-- well if we multiply both of these times 10, that's the exact same thing as 92 divided by 115. We're essentially moved the decimal to the right for both of them. And let me add some zeros here because I suspect that I'm going to get a decimal here. So let's think what this is going to be. Let's think about this. Well 115 doesn't go into 9. It doesn't go into 92. It does go into 920. And I'm going to eyeball and say that it will go eight times. Let's see if that works out. So I have my decimal here. That's a 0. 8 times-- 8 times 5 is 40. 8 times 11 is 88. And then 88 plus 4 is 92. Oh, it went in exactly, very good. So 920, we have no remainder. So 9.2 divided by 11.5 simplified to 0.80. And then 10 to the fifth divided by 10 to the second, we have the same base, and we're dividing. So we can subtract the exponents. That's going to be 10 to the 5 minus 2. So this right over here is going to be 10 to the third power-- so times 10 to the third power. Now, are we done? Well in order to be done, this number right over here needs to be greater than or equal to 1 and less than 10. It is clearly not greater than or equal to 1. So how can we rewrite this as the product of something that is greater than or equal to 1 and less than 10 and some power of 10? Well this 8 right over here, this is in the tenths place. It's 8/10, 8 times 1/10. So this is going to be the same thing as 8 times 10 to the negative 1 power. And then we have this 10 to the third here-- so times 10 to the third power. We'll do that in that other color. And now we have the same base. Just add the exponents. So this is going to be equal to 8 times 10 to the 3 minus 1-- so 8 times 10 squared. And we're done. We've simplified our original expression.