Addition and Subtraction Operations: Videos & Practice Problems

When working with scientific notation, it is essential to ensure that the exponents are the same when adding or subtracting values. The general form for these operations can be expressed as \( a \times 10^x \) and \( b \times 10^x \). In this case, the coefficients \( a \) and \( b \) will be added or subtracted, while the exponent \( x \) remains unchanged. For example, if you are subtracting, the expression simplifies to \( (a - b) \times 10^x \), and for addition, it becomes \( (a + b) \times 10^x \).

However, if the exponents differ, adjustments must be made to align them. For instance, if you have \( 10^8 \) and \( 10^5 \), you would convert \( 10^5 \) to \( 10^8 \) by adjusting the coefficient accordingly. This transformation allows both values to share the same exponent, enabling you to perform the addition or subtraction.

It is also crucial to remember that when adding or subtracting, the final result should reflect the least number of decimal places found in the coefficients. Conversely, when multiplying or dividing, the result should be expressed with the least number of significant figures. Understanding these rules ensures accuracy in calculations involving scientific notation.

As you practice these operations, consider attempting example problems independently. If you encounter difficulties, reviewing different approaches can enhance your understanding of the concepts involved.

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 facilitate the addition, we need to convert \(8.17 \times 10^8\) to the same exponent as \(1.25 \times 10^9\). This involves increasing 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 \(0.817\), leading to the expression \(0.817 \times 10^9\).

Now, we can add the two terms together, keeping the exponent constant:

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

Calculating the sum of the coefficients gives us:

\(0.817 + 1.25 = 2.067\)

Thus, we have:

\(2.067 \times 10^9\)

However, when adding coefficients, we must consider the number of decimal places in the original values. The coefficient \(0.817\) has three decimal places, while \(1.25\) has two. Therefore, our final answer should be rounded to the least number of decimal places, which is two. Rounding \(2.067\) gives us \(2.07\).

Consequently, the final answer is:

\(2.07 \times 10^9\)

To solve problems involving scientific notation, it is essential to ensure that all values share the same exponent before performing any arithmetic operations. In this case, we start with three values, with the largest exponent being \(10^{-11}\). The goal is to convert the other values to this exponent for consistency.

First, we maintain the value \(10^{-11}\) as it is. Next, we address the value \(10^{-12}\). To convert this to \(10^{-11}\), we need to increase the exponent by 1. This requires us to adjust the coefficient by moving the decimal point one place to the left, resulting in:

\( -0.117 \times 10^{-11} \)

For the value \(10^{-13}\), we need to increase the exponent by 2. This means moving the decimal point two places to the left, yielding:

\( 0.035 \times 10^{-11} \)

Now that all values are expressed with the same exponent of \(10^{-11}\), we can perform the subtraction. The operation involves subtracting the adjusted coefficients:

\( 10^{-11} - (-0.117 \times 10^{-11}) - (0.035 \times 10^{-11}) \)

Calculating this gives:

\( 8.9295 \times 10^{-11} \)

When adding or subtracting numbers in scientific notation, it is crucial to consider the precision of the values involved. The least number of decimal places among the coefficients dictates the precision of the final answer. In this case, the coefficient with the least decimal places has 2 decimal places. Therefore, we round the result to:

\( 8.93 \times 10^{-11} \)

This final value represents the correct answer after converting all scientific notation values to the same exponent and performing the necessary arithmetic operations.