Addition and Subtraction Operations: Videos & Practice Problems

When performing operations with numbers in scientific notation, it is crucial that the exponents match to add or subtract the coefficients effectively. For instance, to add

$8.17\times {10}^8$

and

$1.25\times {10}^9$

, you must adjust the smaller exponent to match the larger one, ensuring that the coefficients have the least number of decimal places for accuracy. This involves shifting the decimal point of the coefficient in the opposite direction of the exponent change. In multiplication and division, however, focus on the least number of significant figures. For subtraction, like

$8.9295\times {10}^{-11}$

minus values with different exponents, convert all to the same exponent, then subtract while maintaining the least decimal places in the result. Remember, the reciprocal relationship between the coefficient and exponent is key: altering one requires an opposite change to the other.

Scientific Notation in Addition and Substraction

So here, when you add or subtract values in scientific notation, they must have the same exponents. So the coefficients will add or subtract depending on what operation is going on, but the exponents remain constant. So here we have 8×10X -, b×10X. Again, both of them must have the same exponent, so the same power, in order to add or subtract with one another. So here this would just become a-b×10X, and then if we're adding them it would be A+B×10X.

Now there will come situations where the exponents do not match. What do we do in those situations? We're going to save the exponents are not the same that we transform the smaller value so that they do. So let's say that you had one value that was ×108, and the other one was ×105. You'd have to increase the 105 to 108 so that both of them would have the same exponent. Then we could either subtract or add them to together.

Now it says Remember when adding or subtracting values that the final answer must have the least number of decimal places. So when it comes to adding or attracting, we want least number of decimal places in our coefficient when we're multiplying or dividing its least number of significant figures. Now that we've learned the basics in terms of this in these operations, let's take a look at example one. You can attempt to try to do it on your own. If you get stuck, come back and take a look at my video and see how I approach this very question.

Scientific Notation Addition

Using the method discussed above, determine the answer to the following question. So here it's 8.17×108 + 1.25×109. 109 is the larger power, so that's the larger value. So that would mean that we need to convert the smaller value to match that same exponent.

So here we have 108:00, so we need to increase it by one. Now if we're trying to increase the exponent by 1, so we're going to increase by 1. That means that we're going to have to decrease your coefficient by 1 decimal place. So remember, there is a reciprocal or opposite relationship between your coefficient and your exponent. Whatever happens to one, the opposite happens to the other.

So we're going to need to make this number, this coefficient value smaller so that my power of eight can increase to become the power of nine. So I'm going to take this decimal. I'm going to move it over 1 so that we go from 8.17 to 0.817, and by baking that smaller, this just became larger, so plus 1.25×109. Now that both of them have the same exact exponent, I can finally add them together.

The exponent stays constant and all I'm doing now is I'm adding 0.817 plus 1.25. So when I add those two together it gives me 2.067×109. But remember, when we're adding or subtracting coefficients, we want the least number of decimal places. So for the first value we have 3 decimal places. Remember, decimal places are the numbers to the right of the decimal point. This has three decimal places. This here has 2 decimal places.

So my answer at the end must have two decimal places total. So we're going to have to round this to 2.07 and it'll be ×109. So that would be my final answer here. Follow the rules that we observed up above.

Scientific Notation Subtraction

Using the method discussed above, determine the answer to the following question. All right, so remember we're going to go with the largest value. The largest value we have here is 10-11. That would mean my other two values have to be converted to their exponents. Also become ten to the -11, so this stays the same, so we're going to bring it over.

Now for the next number, it's 10-12, so I need to need it to increase by 1:00 so that it matches 10-11. If I want to increase it by one, that means I need to make my coefficient smaller by 1 decimal place so I can move this decimal over here. So it's going to become -0.117×10-11. This one is to the 10-13. I also needed to get it to 10-11, but now I'm going to have to move it 2 decimal places, so it's going to go 1/2. So it's going to become 0.0335×10-11.

Now that all the exponents are 10-11, that just comes down and stays constant. So it's going to be this minus this minus this to give me 8.9295. But remember when it comes to adding and subtracting, we want the least number of decimal places. So here this one here has 2 decimal places, this one here has three decimal places, this one here has four decimal places. So you want least number of decimal places. So we need 2. So it's going to become 8.93×10-11 as my final answer.

So that'll be my answer when I'm converting all of my scientific notation values to the same exponents and then subtracting them from one another.

Here’s what students ask on this topic:

How do you add numbers in scientific notation with different exponents?

To add numbers in scientific notation with different exponents, you must first adjust the exponents to be the same. For example, to add 8.17 × 10⁸ and 1.25 × 10⁹, convert the smaller exponent to match the larger one. This involves shifting the decimal point of the coefficient in the opposite direction of the exponent change. In this case, 8.17 × 10⁸ becomes 0.817 × 10⁹. Then, add the coefficients: 0.817 + 1.25 = 2.067. Finally, ensure the result has the least number of decimal places, giving 2.07 × 10⁹.

What is the rule for decimal places when adding or subtracting in scientific notation?

When adding or subtracting numbers in scientific notation, the final answer must have the least number of decimal places present in the original coefficients. For example, if you add 8.17 × 10⁸ (3 decimal places) and 1.25 × 10⁹ (2 decimal places), the result should be rounded to 2 decimal places, giving 2.07 × 10⁹.

How do you subtract numbers in scientific notation with different exponents?

To subtract numbers in scientific notation with different exponents, first convert all exponents to be the same. For example, to subtract 8.9295 × 10^-11 from values with different exponents, adjust the exponents to match 10^-11. Convert the other values accordingly, then subtract the coefficients. Ensure the result has the least number of decimal places. For instance, if the coefficients have 2, 3, and 4 decimal places, the final answer should have 2 decimal places.

What is the reciprocal relationship between the coefficient and exponent in scientific notation?

The reciprocal relationship between the coefficient and exponent in scientific notation means that any change in the exponent requires an opposite change in the coefficient. For example, if you increase the exponent by 1, you must decrease the coefficient by moving the decimal point one place to the left. Conversely, if you decrease the exponent by 1, you must increase the coefficient by moving the decimal point one place to the right.

How do you handle significant figures in multiplication and division of scientific notation?

When multiplying or dividing numbers in scientific notation, the final answer must have the least number of significant figures present in the original numbers. For example, if you multiply 3.45 × 10² (3 significant figures) by 2.1 × 10³ (2 significant figures), the result should be rounded to 2 significant figures, giving 7.2 × 10⁵.