Addition and Subtraction Operations - Online Tutor, Practice Problems & Exam Prep

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.

Addition and Subtraction Operations

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.

Addition and Subtraction Calculations 1

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.

Addition and Subtraction Calculations 2

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.