Addition and Subtraction Operations: Videos & Practice Problems

Scientific Notation Addition

To solve the expression \(8.17 \times 10^8 + 1.25 \times 10^9\), we first identify that \(10^9\) is the larger power. To add these two numbers, we need to express both terms with the same exponent. Since \(10^8\) is smaller, we will convert it to match \(10^9\).

To do this, we increase the exponent of \(10^8\) by 1, which requires us to decrease the coefficient by moving the decimal point one place to the left. Thus, \(8.17\) becomes \(0.817\). Now we can rewrite the expression as:

\(0.817 \times 10^9 + 1.25 \times 10^9\)

With both terms now having the same exponent, we can add the coefficients:

\(0.817 + 1.25 = 2.067\)

This gives us:

\(2.067 \times 10^9\)

However, when adding or subtracting coefficients, the result should reflect the least number of decimal places from the original values. The coefficient \(0.817\) has three decimal places, while \(1.25\) has two. Therefore, we round \(2.067\) to two decimal places, resulting in:

\(2.07 \times 10^9\)

This is the final answer, adhering to the rules of significant figures and proper scientific notation.

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, first identify the larger exponent. Then, convert the smaller exponent to match the larger one by shifting the decimal point of its coefficient to the left, decreasing the coefficient accordingly. This maintains the reciprocal relationship between the coefficient and exponent. Once both numbers have the same exponent, add the coefficients together while keeping the exponent constant. Finally, round the result to the least number of decimal places found in the original coefficients to ensure accuracy. For example, to add ${8.17}^8$ and ${1.25}^9$, convert ${8.17}^8$ to ${0.817}^9$, then add coefficients: 0.817 + 1.25 = 2.067, and round to two decimal places to get ${2.07}^9$.

What is the rule for subtracting numbers in scientific notation?

When subtracting numbers in scientific notation, ensure all numbers have the same exponent by converting the smaller exponents to match the largest one. Adjust the coefficients by shifting their decimal points to the left, decreasing their values accordingly. After conversion, subtract the coefficients while keeping the exponent constant. The final answer should be rounded to the least number of decimal places present in the original coefficients. For example, subtracting ${8.9295}^{-11}$ minus values with exponents ${10}^{-12}$ and ${10}^{-13}$ requires converting all to ${10}^{-11}$, then subtracting coefficients and rounding to two decimal places, resulting in ${8.93}^{-11}$.

Why must exponents be the same when adding or subtracting numbers in scientific notation?

Exponents must be the same when adding or subtracting numbers in scientific notation because the operation applies only to the coefficients, which represent the significant digits. Scientific notation expresses numbers as a coefficient multiplied by 10 raised to an exponent. If exponents differ, the numbers represent different scales, making direct addition or subtraction invalid. By converting all numbers to the same exponent, you align their scales, allowing you to add or subtract the coefficients correctly. This ensures the mathematical integrity of the operation and accurate results.

How do you determine the number of decimal places to keep in the result of addition or subtraction in scientific notation?

When adding or subtracting numbers in scientific notation, the result should be rounded to the least number of decimal places found in the original coefficients. Decimal places refer to the digits to the right of the decimal point. For example, if one coefficient has three decimal places and another has two, the final answer must be rounded to two decimal places. This rule ensures that the precision of the result does not exceed the least precise measurement, maintaining consistency and accuracy in calculations.

What is the relationship between the coefficient and exponent when adjusting numbers in scientific notation?

There is a reciprocal relationship between the coefficient and exponent in scientific notation. When you increase the exponent by one, you must decrease the coefficient by shifting its decimal point one place to the left, making it smaller. Conversely, decreasing the exponent by one requires increasing the coefficient by shifting its decimal point one place to the right. This balance keeps the overall value of the number constant. For example, converting ${8.17}^8$ to an exponent of 9 involves changing the coefficient to $0.817$ and the exponent to 9.