Problem 13 - Chemistry Matter and Change

Question

Solve each problem, and express the answer in scientific notation. a. \(\left(5 \times 10^{-5}\right)+\left(2 \times 10^{-5}\right) \quad\) c. \(\left(9 \times 10^{2}\right)-\left(7 \times 10^{2}\right)\) b. \(\left(7 \times 10^{8}\right)-\left(4 \times 10^{8}\right) \quad\) d. \(\left(4 \times 10^{-12}\right)+\left(1 \times 10^{-12}\right)\)

Step-by-Step Solution

Answer

a. \(7 \times 10^{-5}\) b. \(3 \times 10^{8}\) c. \(2 \times 10^{2}\) d. \(5 \times 10^{-12}\)

Step 1: Ensure the exponents are the same

The given numbers already have the same exponents, so we can proceed to the next step. \( \left(5 \times 10^{-5}\right)+\left(2 \times 10^{-5}\right) \)

Step 2: Add the coefficients.

Add the coefficients while keeping the exponent the same: \( \left(5 + 2\right) \times 10^{-5} \)

Step 3: Write the result in scientific notation.

Combine the coefficients and write the result in scientific notation: \( 7 \times 10^{-5} \) #b. Subtract the numbers in scientific notation#

Step 1: Ensure the exponents are the same

The given numbers have the same exponent, so we can proceed to the next step. \( \left(7 \times 10^{8}\right)-\left(4 \times 10^{8}\right) \)

Step 2: Subtract the coefficients.

Subtract the coefficients keeping the exponent the same: \( \left(7 - 4\right) \times 10^{8} \)

Step 3: Write the result in scientific notation.

Combine the coefficients and write the result in scientific notation: \( 3 \times 10^{8} \) #c. Subtract the numbers in scientific notation#

Step 1: Ensure the exponents are the same

The given numbers have the same exponent, so we can proceed to the next step. \( \left(9 \times 10^{2}\right)-\left(7 \times 10^{2}\right) \)

Step 2: Subtract the coefficients.

Subtract the coefficients keeping the exponent the same: \( \left(9 - 7\right) \times 10^{2} \)

Step 3: Write the result in scientific notation.

Combine the coefficients and write the result in scientific notation: \( 2 \times 10^{2} \) #d. Add the numbers in scientific notation#

Step 1: Ensure the exponents are the same

The given numbers already have the same exponents, so we can proceed to the next step. \( \left(4 \times 10^{-12}\right)+\left(1 \times 10^{-12}\right) \)

Step 2: Add the coefficients.

Add the coefficients while keeping the exponent the same: \( \left(4 + 1\right) \times 10^{-12} \)

Step 3: Write the result in scientific notation.

Combine the coefficients and write the result in scientific notation: \( 5 \times 10^{-12} \)

Key Concepts

Addition and Subtraction in Scientific Notation

When you're dealing with numbers in scientific notation, addition and subtraction might seem tricky at first, but they're actually quite straightforward once you understand the process. Scientific notation is a way to express very large or very small numbers in the form \(a \times 10^n\), where \(a\) is the coefficient, and \(n\) is the exponent of 10. Here’s how you can easily add or subtract such numbers.

• Ensure the Exponents Match: Before adding or subtracting, check if the numbers have the same exponent. If not, you’ll need to adjust one of them so they match. This step is crucial because you can only combine numbers directly when their place values align, akin to ensuring the decimal points are aligned in standard addition and subtraction.


• Add or Subtract the Coefficients: With matching exponents, the next step is simple. Add or subtract the coefficients (\(a\) values) as dictated by the problem. For example, given \((5 \times 10^{-5}) + (2 \times 10^{-5})\), you simply add the 5 and 2 to get 7, giving you \(7 \times 10^{-5}\).


• Recombine with the Unchanged Exponent: Once added or subtracted, your new coefficient is recombined with the unchanged exponent to complete the scientific notation expression.

Master these simple steps, and you’ll find that adding and subtracting in scientific notation becomes second nature!

Exponents

Exponents are interesting little numbers that tell us how many times to multiply a number by itself. In scientific notation, exponents are key because they let us express really big or really tiny numbers succinctly.

• Understanding Exponents: An exponent \(n\) in a term like \(10^n\) tells you how many times you'll multiply 10 by itself. For instance, \(10^3\) is 1000 because 10 is used three times in multiplication: \(10 \times 10 \times 10\).


• Negative Exponents: These indicate division instead of multiplication. For example, \(10^{-2}\) means \(1/10^2\) which is 0.01. This is vital for representing tiny values like 0.00001 as \(1 \times 10^{-5}\).


• Rule Consistency in Operations: When we add or subtract numbers, the exponents involved have to match first. This is because the exponent represents a scale, so for direct combination, we are required to work within the same scale.

Exponents simplify calculations and presentations by minimizing the number of digits we need to write, making complex mathematics more manageable.

Coefficients

The coefficient in scientific notation is the first number in the expression \(a \times 10^n\). It’s an important number because it represents a scaled down version of the original number you’re working with, making it easier to manage and understand.

• What is a Coefficient?: This is the \(a\) part of scientific notation. It's essentially the important part of the original number, stripped of extra zeros thanks to the exponent. If a scientific notation reads \(3.5 \times 10^4\), then 3.5 is your coefficient.


• Using Coefficients in Operations: When adding or subtracting in scientific notation, it is the coefficients that you manipulate. They need to be calculated directly, while the exponent remains constant. This simplifies operations like \((6.1 \times 10^3) + (4.3 \times 10^3)\), where you only add 6.1 and 4.3 to get 10.4.


• Ensuring the Range of the Coefficient: Typically in scientific notation, the coefficient is a number between 1 and 10, ensuring clarity and consistency.

The coefficient acts as the heart of the scientific notation, giving the expression its precise and readable magnitude.