Problem 28 - Elementary Algebra

Question

\(\frac{8 \times 10^{9}}{2 \times 10^{-3}}\)

Step-by-Step Solution

Answer

4 \times 10^{12}

Step 1: Rewrite the Exponent

Rewrite the given expression using the properties of exponents. This can help in simplifying the calculation. Given expression: \[ \frac{8 \times 10^{9}}{2 \times 10^{-3}} \]

Step 2: Divide the Coefficients

Divide the coefficients (8 and 2) separately from the powers of ten. This gives: \[ \frac{8}{2} \times \frac{10^{9}}{10^{-3}} = 4 \times \frac{10^{9}}{10^{-3}} \]

Step 3: Apply the Rule of Exponents

Use the properties of exponents \( \frac{a^m}{a^n} = a^{m-n} \) to simplify \( \frac{10^{9}}{10^{-3}} \): \[ 10^{9 - (-3)} = 10^{9+3} = 10^{12} \]

Step 4: Combine the Results

Combine the simplified coefficient with the simplified power of ten: \[ 4 \times 10^{12} \]

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential when simplifying expressions. Here are some crucial rules:

• Product Rule: When multiplying, add the exponents. That is, \(a^m \times a^n = a^{m+n}\).
• Quotient Rule: When dividing, subtract the exponents. That is, \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent Rule: Any number raised to the power 0 is 1. That is, \(a^0 = 1\).
• Negative Exponent Rule: A negative exponent indicates the reciprocal. That is, \(a^{-n} = \frac{1}{a^n}\).

In the given exercise, these rules help us simplify the expression more effectively. By understanding and applying them, calculations become more straightforward and manageable.

Dividing Coefficients

When simplifying expressions involving exponents, it's important to handle the coefficients separately from the powers of ten. In the provided example:

\[ \frac{8 \times 10^{9}}{2 \times 10^{-3}} \]

We start by dividing the coefficients 8 and 2:

\[ \frac{8}{2} = 4 \]

This step simplifies the coefficient part of the expression. Always separate coefficients and exponents before applying further operations.

By dividing coefficients first, the entire process becomes more structured and prevents errors in simplifying.

Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently.

• A number in scientific notation is written as \(a \times 10^n\), where 1 ≤ a < 10 and n is an integer.
• It simplifies arithmetic operations, especially multiplication and division involving powers of ten.
• Scientific notation makes it easier to read, write, and compute with large or small quantities in science and engineering.

In our example expression, \( 4 \times 10^{12} \):

• 4 is the coefficient, a manageable number between 1 and 10.
• 10^{12} indicates that the actual number is 4 followed by 12 zeros (or 4,000,000,000,000).

By understanding and using scientific notation, we simplify complex figures and make calculations clearer and more efficient.