Problem 89 - College Algebra

Question

Write each number in scientific notation and use scientific notation to perform the operation(s). Express the answer in scientific notation. $$ \frac{480,000,000,000}{0.00012} $$

Step-by-Step Solution

Answer

The result is \(4 \times 10^{15}\)

Step 1: Convert Numbers into Scientific Notation

The numbers will be represented in scientific notation, where a decimal number is expressed in terms of a number between 1 and 10 multiplied by a power of 10. Therefore, \[ 480,000,000,000 = 4.8 \times 10^{11} \] and \[ 0.00012 = 1.2 \times 10^{-4} \] The original expression becomes \[ \frac{4.8 \times 10^{11}}{1.2 \times 10^{-4}} \]

Step 2: Perform the Division Operation Using the Rules of Exponents

We can split the fraction into two by dividing the coefficients (the numbers) and then the powers of 10 separately. So, \[ \frac{4.8}{1.2}=4 \] and \[ \frac{10^{11}}{10^{-4}}=10^{11-(-4)}=10^{15} \] Multiplying these two results, we get \[ 4 \times 10^{15} \] this is the result of the division operation.

Step 3: Write the Final Answer in Scientific Notation

The final answer already is in scientific notation: \[ 4 \times 10^{15} \] Therefore, there is nothing further to do in this step.

Key Concepts

Understanding the Rules of Exponents

Exponents are a way to express repeated multiplication. When dealing with scientific notation, the rules of exponents help simplify expressions. These rules include:

• Product of Powers: When multiplying like bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).

These rules allow us to manage complex expressions more easily, especially when working with powers of 10 in scientific notation. Mastery of these rules simplifies calculations and aids in maintaining accurate representations.

Simplifying with Division in Scientific Notation

The division operation is vital when working with scientific notation. To perform division, split the problem into two parts:

• Divide the numerical coefficients.
• Apply the quotient of powers rule to divide the powers of 10.

For example, consider: \( \frac{4.8 \times 10^{11}}{1.2 \times 10^{-4}} \). First, divide the coefficients: \( \frac{4.8}{1.2} = 4 \).
Then subtract the exponents: \( 11 - (-4) = 15 \), giving \( 10^{15} \).
This results in \( 4 \times 10^{15} \). Breaking down the division in this way helps maintain clarity and precision.

Exploring Powers of 10

Powers of 10 are foundational to scientific notation. They represent large or small numbers compactly:

• Positive exponents (e.g., \( 10^3 \)) mean moving the decimal right (1000).
• Negative exponents (e.g., \( 10^{-2} \)) mean moving the decimal left (0.01).

This system allows for efficient expression and computation and helps with understanding scale and magnitude in various scientific fields. Knowing how to manipulate powers of 10 is crucial for scientific computations, enabling easy comparison and adjustment of large and small values.

Scientific Notation as Educational Material

Scientific notation is a key tool in both education and professional scientific work. It teaches students how to handle real-world data efficiently:

• Simplification: Makes complex numbers manageable.
• Clarity: Provides clear representation of very large or small numbers.
• Application: Useful in fields like astronomy, engineering, and physics.

Using scientific notation helps in building critical skills in estimating, calculating, and interpreting numerical data. Its educational benefits extend beyond mathematics, emphasizing the importance of scalability and precision.