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

When working with scientific notation, it is essential to ensure that the exponents are the same before performing addition or subtraction. In this context, the coefficients can be added or subtracted, while the exponent remains unchanged. For example, if you have two values expressed as \( a \times 10^x \) and \( b \times 10^x \), the operation would yield \( (a \pm b) \times 10^x \).

However, if the exponents differ, adjustments must be made to align them. For instance, if one value is \( a \times 10^8 \) and the other is \( b \times 10^5 \), you would convert \( b \times 10^5 \) to \( b \times 10^8 \) 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 present in the coefficients. Conversely, when multiplying or dividing, the result should be expressed with the least number of significant figures. This distinction is vital for maintaining precision in calculations.

By mastering these principles, you can confidently navigate operations involving scientific notation, ensuring accuracy and clarity in your mathematical expressions.

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.

Scientific Notation Subtraction

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:

\( 0.0000000000 - 0.117 + 0.035 = 8.9295 \)

When adding or subtracting numbers in scientific notation, it is crucial to consider the precision of the values involved. The original coefficients had 2, 3, and 4 decimal places, respectively. Therefore, we round the final result to the least number of decimal places, which is 2 in this case. Thus, the final answer is:

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

This method highlights the importance of aligning exponents and maintaining precision in scientific calculations.

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.