Addition and Subtraction Operations: Study with Video Lessons, Practice Problems & Examples

When adding or subtracting values in scientific notation, ensure the exponents are the same. Adjust coefficients accordingly, noting that increasing the exponent decreases the coefficient by one decimal place. For example, to add \(8.17 \times 10^8\) and \(1.25 \times 10^9\), convert \(8.17 \times 10^8\) to \(0.817 \times 10^9\) before adding. The final answer should reflect the least number of decimal places from the coefficients involved. This principle applies to all operations in scientific notation, emphasizing precision in significant figures.

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 a×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 are adding them it would be again ×10x. Now there will come situations where the exponents do not match, what do we do in those situations? We are going to say if the exponents are not the same then 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 would have to increase the 105 to 108 so that both of them would have the same exponent. Then we can either subtract or add them together.

Now it says remember when adding or subtracting values that the final answer must have the least number of decimal places. When it comes to adding or subtracting, we want the least number of decimal places in our coefficient. When we are multiplying or dividing it is the least number of significant figures. Now that we have learned the basics in terms of these operations, let's take a look at example 1. 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 is 8.17 times 10 to the 8 plus 1.25x10⁹. 10 to the 9 is the larger power, so that is the larger value. That would mean we need to convert the smaller value to match that same exponent. So here we have 10 to the 8 so we need to increase it by 1. 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 1, the opposite happens to the other. So we're gonna need to make this number this coefficient value smaller so that my power of 8 can increase to become the power of 9. So I'm gonna take this decimal, I am going to move it over 1 so that we go from 8.17 to 0.817. And by making that smaller this just became larger. So plus 1.25 times 10⁹. Now that both of them have the same exact exponent I can finally add them together. The exponent stays constant, and all I am doing now is adding 0.817 plus 1.25. So when I add those 2 together it gives me 2.067 times 10⁹. But remember when we are 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 2 decimal places total. So we are going to have to round this to 2.07 and it will be times 10⁹. That would be my final answer here, following the rules that we observed up above.

Addition and Subtraction Calculations 2

Using the method discussed above, determine the answer to the following question. Alright, so remember we are 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 so that their exponents also become 10-11. So this stays the same so we are going to bring it over. Now for the next number, it is 10-12, so I need it to increase by 1 so that it matches 10-11. If I want to increase it by 1, that means I need to make my coefficient smaller by 1 decimal place. So I am going to move this decimal over here. So it is going to become negative 0.117 times 10-11. This one is to the negative 13; I also need to get it to 10-11, but now I am going to have to move it 2 decimal places. So it is going to go 12 so It is going to become 0.035 times 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 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 3 decimal places, and this one here has 4 decimal places. So we want the least number of decimal places, so we need 2. So it is going to become 8.93 times 10-11 as my final answer. So that will be my answer when I am 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 or subtract numbers in scientific notation with different exponents?

To add or subtract numbers in scientific notation with different exponents, you must first adjust the exponents to be the same. This involves transforming the smaller exponent to match the larger one. You do this by moving the decimal point of the coefficient in the opposite direction. For example, if you have 8.17 x 10⁸ and 1.25 x 10⁹, you adjust 8.17 x 10⁸ to 0.817 x 10⁹. Once the exponents are the same, you can add or subtract the coefficients while keeping the exponent constant. Finally, ensure the result has the least number of decimal places from the coefficients involved.

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

When adding or subtracting numbers in scientific notation, the final result should have the least number of decimal places from the coefficients involved in the calculation. This means you look at the number of decimal places in each coefficient and use the smallest number of decimal places for your final answer. For instance, if you add 0.817 (3 decimal places) and 1.25 (2 decimal places), your result should be rounded to 2 decimal places, such as 2.07 x 10⁹.

Why is it important to have the same exponent when adding or subtracting in scientific notation?

Having the same exponent when adding or subtracting in scientific notation is crucial because it ensures that the numbers are on the same scale, allowing for accurate arithmetic operations. If the exponents differ, the values represent different magnitudes, making direct addition or subtraction incorrect. By adjusting the exponents to be the same, you align the numbers to a common scale, enabling you to correctly add or subtract the coefficients while keeping the exponent constant.

Can you explain the relationship between coefficients and exponents in scientific notation?

In scientific notation, there is an inverse relationship between coefficients and exponents. When you adjust the exponent, you must move the decimal point of the coefficient in the opposite direction to maintain the value of the number. For example, if you increase the exponent by 1, you decrease the coefficient by moving the decimal point one place to the left. This reciprocal adjustment ensures that the overall value of the number remains unchanged while aligning exponents for addition or subtraction.

How do you handle negative exponents when adding or subtracting in scientific notation?

When dealing with negative exponents in scientific notation, the process is similar to positive exponents. You must ensure all numbers have the same exponent before performing addition or subtraction. For example, if you have numbers with exponents of 10^-11, 10^-12, and 10^-13, adjust the smaller exponents to match the largest one, 10^-11. This involves moving the decimal point of the coefficients to the right, increasing the exponent. Once aligned, you can add or subtract the coefficients, keeping the exponent constant, and ensure the result has the least number of decimal places.