Problem 44 - Astronomy: At Play in the Cosmos

Question

Write a million billion billion in scientific notation.

Step-by-Step Solution

Answer

A million billion billion is written as \(10^{24}\) in scientific notation.

Step 1: Understanding Exponentiation

First, understand what a million, billion, and another billion represent in terms of powers of ten. A million is \(10^6\), a billion is \(10^9\). Thus, a million billion billion can be expressed as \(10^6 \times 10^9 \times 10^9\).

Step 2: Multiplying Exponents

Next, multiply the exponents of ten. According to the Laws of Exponents, when you multiply powers with the same base, you add the exponents. So, \(10^6 \times 10^9 \times 10^9 = 10^{6+9+9} = 10^{24}\).

Step 3: Writing in Scientific Notation

Now, express the result in proper scientific notation. Scientific notation is typically written as a product of a number between 1 and 10, and a power of 10. Here, \(10^{24}\) is already in scientific notation form.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers called the base and the exponent. This operation allows us to express repeated multiplication of a base number. For example,

• The expression \(10^3\) means that the base 10 is multiplied by itself 3 times: \(10 \times 10 \times 10\).
• In general, \(a^n\) implies that you multiply the base \(a\) by itself \(n\) times.

Exponentiation is essential for simplifying calculations and expressing large numbers in a compact form. It provides a more straightforward way to handle numbers with many digits through use of exponents.

Powers of Ten

The powers of ten are a concept applied when dealing with very large or very small numbers. Each power of ten is represented as \(10\) raised to an exponent, indicating how many times to multiply or divide by 10. Here are some key examples:

• \(10^0 = 1\), as any number to the zero power is one.
• \(10^1 = 10\).
• \(10^2 = 100\), suggesting a multiplication of 10 by itself once.
• \(10^3 = 1000\), reflecting a multiplication by 10 two more times.

In scientific notation, powers of ten enable easy expression of extreme values, like a million billion billion, compactly as \(10^{24}\). This technique efficiently communicates very large numbers while reducing complexity.

Laws of Exponents

The laws of exponents, also known as the rules of exponents, simplify calculations involving exponential expressions. Key rules include:

• Product of Powers Rule: When multiplying numbers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When raising a power to another exponent, you multiply the exponents: \((a^m)^n = a^{m \times n}\).

These laws help to break down complex expressions into manageable pieces, ensuring that computations are both efficient and accurate. For example, using the product of powers rule allows us to find \(10^6 \times 10^9 \times 10^9\) by simply adding exponents to get \(10^{24}\).