Evaluate 12000000000 times 10000000 over two hundredths times 20000000 times 500000, expressing it in scientific notation.
I wanna show you two methods. The first one, we’ll simplify before we put this into scientific notation. And in the second method, we’ll put everything in scientific notation and then we’ll simplify. To simplify, what we’ll do is take the 12 and the one in the numerator and keep that. 12 times one equals 12. And there are 16 zeros remaining. And that means we need a one followed by 16 zeros, like this.
In the denominator, we have two times five. But that equals 10, which means it’s just another zero. For now, we won’t deal with this 0.02. We’ll just bring it down. And that is being multiplied by a one followed by 13 zeros. And we know that the 13 zeros on the bottom and the 13 zeros on the top can cancel out.
We now have 12000 in the numerator and two hundredths in the denominator. To get the two hundredths out of the denominator, I wanna move the decimal two places to the right. And I can do that by multiplying by 100. But if I multiply the denominator by 100, I need to multiply the numerator by 100. And that means our numerator will be a 12 followed by five zeros, and our denominator will be two. We can then divide 12 by two, which equals six. And we’ll bring down our five zeros.
This expression is now completely simplified. It needs to be put into scientific notation. Scientific notation requires that the ones place be between zero and nine. So we would rewrite this as six multiplied by 10 to some power. Well, we need to move the decimal five places to the right. And that means we’ll need 10 to the fifth power. This fraction in scientific notation is six times 10 to the fifth power.
But what if we wanted to put everything in scientific notation before we multiplied and simplified? We’ll start with the same expression. And then we’ll write 12000000000 in scientific notation. That would be 1.2 times 10 to the 10th power. Next, 10,000,000 in scientific notation would be 1.0 times 10 to the seventh power.
Moving into the denominator, how would we write two hundredths in scientific notation? 2.0 times 10 to the negative two power. 20000000 becomes 2.0 times 10 to the seventh power. And then 500000 is 5.0 times 10 to the fifth power. We notice that there is a 10 to the seventh in the numerator and the denominator. Those can cancel out. Our 10 to the fifth in the denominator will cancel, leaving 10 to the fifth in the numerator.
Our numerator is now 1.2 times 10 to the fifth, and then our denominator two times 10 to the negative two times two times five. Two times five is 10, or you could write it as 10 to the first power. We can do some more cancelling. This 10 to the first power makes the numerator 10 to the fourth power.
And then I wanna break these two pieces up. What is 1.2 divided by two? And then we’ll multiply 10 to the fourth over 10 to the negative two. 1.2 divided by two or half of 1.2 is 0.6. And when we have a negative exponent in the denominator, it moves to the numerator. So we’ll have 10 to the fourth plus two power 10 to the sixth power.
However, 0.6 is not in scientific notation. We can write 0.6 as six times 10 to the negative one power times 10 to the sixth power. Multiplying exponents with the same base means you add the values of the exponents. 6.0 times 10 to the negative one plus six, 6.0 times 10 to the fifth. Both of these methods are completely accurate. And one of them just saved you from writing so many zeros.