Problem 15 - Chemistry Matter and Change

Question

PRACTICE Problems Solve each problem, and express the answer in scientific notation. a. \(\left(4 \times 10^{2}\right) \times\left(1 \times 10^{8}\right) \quad\) c. \(\left(6 \times 10^{2}\right) \div\left(2 \times 10^{1}\right)\) b. \(\left(2 \times 10^{-4}\right) \times\left(3 \times 10^{2}\right) \quad\) d. \(\left(8 \times 10^{4}\right) \div\left(4 \times 10^{1}\right)\)

Step-by-Step Solution

Answer

a. \(4 \times 10^{10}\), b. \(6 \times 10^{-2}\), c. \(3 \times 10^{1}\), d. \(2 \times 10^{3}\).

Step 1: Multiply Base Numbers for a

First, multiply the base numbers without the powers of 10. That is, calculate \( 4 \times 1 = 4 \).

Step 2: Add Exponents for a

With the base numbers multiplied, add the exponents of 10. Here, \( 2 + 8 = 10 \).

Step 3: Write the Answer for a in Scientific Notation

Combine the results from Steps 1 and 2. So, the problem \((4 \times 10^{2}) \times (1 \times 10^{8})\) becomes \(4 \times 10^{10}\).

Step 4: Multiply Base Numbers for b

First, calculate \( 2 \times 3 = 6 \).

Step 5: Add Exponents for b

Add \(-4 + 2 = -2 \).

Step 6: Write the Answer for b in Scientific Notation

Combine the results to get \( 6 \times 10^{-2} \).

Step 7: Divide Base Numbers for c

Calculate \( 6 \div 2 = 3 \).

Step 8: Subtract Exponents for c

Subtract the exponents \( 2 - 1 = 1 \).

Step 9: Write the Answer for c in Scientific Notation

The answer is \( 3 \times 10^{1} \).

Step 10: Divide Base Numbers for d

Calculate \( 8 \div 4 = 2 \).

Step 11: Subtract Exponents for d

Subtract the exponents \( 4 - 1 = 3 \).

Step 12: Write the Answer for d in Scientific Notation

The answer is \( 2 \times 10^{3} \).

Key Concepts

Multiplication of Exponents

When you are faced with multiplying numbers in scientific notation, focus on two components: the numerical base and the power of ten. This approach breaks down a potentially complex multiplication into simpler steps:

• Multiply the Base Numbers: Take the numerical parts of the numbers and multiply them. For instance, if you are multiplying \((4 \times 10^2)\) and \((1 \times 10^8)\), focus first on the numbers 4 and 1. Simply calculate 4 times 1, which equals 4.


• Add the Exponents: Next, work with the exponents. Add the exponents of the powers of ten. In the example with \((10^2)\) and \((10^8)\), add 2 and 8 to get 10. This step combines the powers of ten, effectively merging them into a new exponent.

By handling multiplication in this two-step manner, you can easily arrive at the result in scientific notation: \(4 \times 10^{10}\). This structured approach simplifies multiplication and ensures clarity, particularly when dealing with large scale numbers in scientific contexts. Remember, it's all about breaking it down into manageable pieces!

Division of Exponents

Just like multiplication, division with exponents involves a step-by-step strategy:

• Divide the Base Numbers: Start by dividing the numerical coefficients. For example, dividing \((6 \times 10^2)\) by \((2 \times 10^1)\) focuses on 6 divided by 2, giving a result of 3.


• Subtract the Exponents: With the numbers taken care of, shift your attention to the exponents. For the exponents 2 and 1, subtract: 2 minus 1 equals 1.

You now have separated and simplified the problem into an easy-to-understand format. The scientific notation, after division, is \(3 \times 10^1\). This systematic procedure helps maintain accuracy, especially when working with diverse scientific data or complex algebraic expressions. The key is in consistently using the subtraction of exponents to clarify the division.

Scientific Notation Problems

Scientific notation offers a way to express very large or very small numbers succinctly. It's particularly useful in scientific and engineering contexts. Problems constructed in scientific notation often require you to handle both multiplication and division of numbers with exponents. This approach helps to:

• Bring Order to Chaos: Large numbers like billions or tiny figures like micrograms become more manageable and easier to communicate.


• Encourage Precision: Scientific notation maintains accuracy by clearly showing the size and significance of numbers involved, which is crucial in precise calculations and comparisons.

For example, when multiplying \((2 \times 10^{-4})\) with \((3 \times 10^2)\), you transform a complex problem into one involving simple arithmetic: 2 times 3 equals 6, and for the exponents, add -4 and 2, resulting in \(6 \times 10^{-2}\). By practicing with these types of problems, students develop a stronger grasp of manipulating exponents and a deeper understanding of quantity representation.

Mathematics Education

Building a strong foundation in mathematical concepts like exponents and scientific notation is crucial for students. Mathematics education focuses on developing problem-solving skills and a logical mindset. Here's how learning these concepts fits into the bigger picture:

• Critical Thinking: By engaging with exercises involving exponents and scientific notation, students learn to dissect problems, strategize solutions, and apply learned knowledge.


• Practical Applications: These mathematical techniques are not just theoretical. They apply directly to real-world problems, whether in measuring distances in space or depths in oceanography.

Effective education nurtures curiosity and understanding, fostering skills that extend beyond the classroom into everyday life and professional careers. Getting comfortable with exponents and learning to express numbers in scientific notation opens doors to advanced topics, laying the groundwork for further studies in sciences, engineering, and technology. Emphasizing clarity and comprehension ensures every student gets the most out of their mathematical journey.