Problem 101 - Chemistry Matter and Change

Question

Perform the following operations. (Chapter 2) a. \(10^{7} \times 10^{3}\) b. \(\left(1.4 \times 10^{-3}\right) \times\left(5.1 . \times 10^{-5}\right)\) c. \(\left(2 \times 10^{-3}\right) \times\left(4 \times 10^{5}\right)\)

Step-by-Step Solution

Answer

a. \(10^{10}\) b. \(7.14 \times 10^{-8}\) c. \(8 \times 10^{2}\)

Step 1: a. Multiplying 10^7 and 10^3

To multiply these numbers, we add their exponents: \(10^{7}\times 10^{3} = 10^{7+3} = 10^{10}\). The result is \(10^{10}\).

Step 2: b. Multiplying 1.4 × 10^-3 and 5.1 × 10^-5

First, multiply the coefficients: \(1.4 \times 5.1 = 7.14\). Next, multiply the powers of 10 by adding their exponents: \(10^{-3} \times 10^{-5} = 10^{-3-5} = 10^{-8}\). Now, combine the results: \(7.14 \times 10^{-8}\). The final answer is \(7.14 \times 10^{-8}\).

Step 3: c. Multiplying 2 × 10^-3 and 4 × 10^5

To multiply these numbers, follow the same steps as before. First, multiply the coefficients: \(2 \times 4 = 8\). Next, multiply the powers of 10 by adding their exponents: \(10^{-3} \times 10^{5} = 10^{-3+5} = 10^{2}\). Now, combine the results: \(8 \times 10^{2}\). The final answer is \(8 \times 10^{2}\).

Key Concepts

Multiplying Powers of Ten

Understanding how to multiply powers of ten is crucial, not just in mathematics but also in the world of science where large or miniscule quantities are common. When multiplying powers of ten, we use a rule that greatly simplifies the process: simply add the exponents.

For instance, given the problem of multiplying 10 to the seventh power by 10 to the third power (\(10^{7} \times 10^{3}\)), you can imagine trying to write out all of those zeros, which is complicated and time-consuming. The beauty of the exponent rules is that they make this easy. Since both bases are 10, we just add the exponents to get a new power of ten: \
10^{7+3} = 10^{10}\
This concise method keeps calculations clean and error-free. This principle is a building block for much more complex problems in algebra, physics, and engineering.

Scientific Notation Multiplication

Multiplying numbers in scientific notation isn’t much different from multiplying powers of ten, it just has an extra step. The purpose of scientific notation is to express very large or very small numbers in a more manageable form. The general form is a number between 1 and 10 multiplied by a power of ten.

When multiplying two numbers in scientific notation, you multiply the coefficients (those numbers between 1 and 10) and then follow the rules for multiplying powers of ten. For example, when multiplying \(1.4 \times 10^{-3}\) by \(5.1 \times 10^{-5}\), the coefficients are 1.4 and 5.1. These multiply to 7.14. Then we multiply the powers of ten, using the exponent addition rule we discussed earlier: \
10^{-3} \times 10^{-5} = 10^{-3-5} = 10^{-8}\
Combining these, you get \
7.14 \times 10^{-8}\. This keeps the numbers approachable and easy to work with, especially important in fields like chemistry, where mole calculations often involve very large or very small numbers.

Exponent Addition

In our exploration of multiplying powers of ten and scientific notation, we’ve come across a fundamental mathematical operation known as exponent addition. This operation is straightforward when the bases of the numbers you are multiplying are the same.

In essence, when you multiply two powers with the same base, you add their exponents to get the power of the product: \(a^m \times a^n = a^{m+n}\). This rule holds true for any real number base 'a' and any integers 'm' and 'n'. In sciences like chemistry or physics, exponent addition comes in handy when dealing with exponential growth, radioactive decay, or calculating concentrations. It simplifies the process of working with very large or very small quantities, allowing scientists and students alike to focus on understanding the concepts rather than getting bogged down in complicated arithmetic.

Chemistry Mathematics

In the realm of chemistry, mathematics plays a foundational role in understanding various concepts, from stoichiometry to the measurement of reaction rates. Working with numbers in scientific notation is a daily task for chemists. It allows the expression of a substance’s concentration in moles per liter, the calculation of Avogadro's number, or even the representation of elementary charge in a manageable way.

Real-world Applications

• Calculating dimensional analyzes for converting units.
• Determining molecular weights and using them in concentration calculations.
• Quantifying reaction yields and rates with exponential and logarithmic relationships.

Chemistry heavily relies on mathematics to ensure precision and accuracy in experimental science. Understanding the concepts behind multiplying powers of ten and scientific notation fosters a deeper comprehension of the significance behind the numbers, be it in the lab or in theoretical models.