Problem 45 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. Aluminum-26 has a half-life of 740,000 years.

Step-by-Step Solution

Answer

Answer: The half-life of Aluminum-26 in scientific notation is 7.4 * 10^5 years.

Step 1: Determine the power of 10

First, determine the power of 10 in scientific notation. Count how many decimal places to move the decimal point so that only one non-zero digit is before the decimal. In this case, we need to move the decimal point 5 places to the left. So, our power of 10 will be 10^5.

Step 2: Rewrite the number as the product of a number between 1 and 10 and a power of 10

Rewrite 740,000 as a product of a number between 1 and 10 and a power of 10. After moving the decimal point, we get 7.4. So, the scientific notation of 740,000 is 7.4 * 10^5. Therefore, the half-life of Aluminum-26 in scientific notation is 7.4 * 10^5 years.

Key Concepts

Understanding Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols to represent numbers and express mathematical relationships. In the context of our exercise, algebra comes into play when we convert a standard number like 740,000 into scientific notation. Scientific notation is an algebraic form that expresses very large or very small numbers succinctly, often used in sciences to make computations more manageable.

For example, when we convert the half-life of Aluminum-26 into scientific notation, we're using algebraic thinking to handle the exponentiation process. Algebra involves understanding how to perform operations on numbers and variables, which are essential when rewriting numbers into a form that involves multiplying by powers of 10.

Exponents Unraveled

Exponents are a critical component of scientific notation and are used to simplify the notation of large numbers. An exponent, also known as a power, is a shorthand way of indicating how many times a number, called the base, is multiplied by itself. In scientific notation, a number is written as the product of a number between 1 and 10 and a power of 10.

Power of Ten in Scientific Notation

In our exercise, we worked with the number 740,000 and moved the decimal point five places to the left to get the number 7.4, which is between 1 and 10. The '5' in 10^5 indicates the number of times we multiply 10 by itself. So, 10^5 (10 to the power of five) means 10 * 10 * 10 * 10 * 10. This use of exponents makes large numbers much easier to work with and understand.

The Concept of Half-Life

Half-life is a term primarily used in nuclear physics and chemistry to describe the time required for half of the unstable atoms in a radioactive sample to undergo radioactive decay. The half-life of a substance affects its duration of radioactivity and can range from fractions of a second to millions of years.

In the case of Aluminum-26 mentioned in our exercise, it has a half-life of 740,000 years, which means that after this period, half of the original amount of Aluminum-26 will have decayed into other elements. Understanding half-life is not only crucial for scientific research but also important in fields like geology (for dating rocks) and medicine (for medical treatments). In scientific notation, the half-life helps us express large time spans in a more compact and computable form, as seen with the notation 7.4 * 10^5 years.