Problem 11 - Algebra 2 and Trigonometry

Question

Write each number as a power. 0.001

Step-by-Step Solution

Answer

0.001 is written as \(10^{-3}\).

Step 1: Identify the Number's Format

The number given, 0.001, can be expressed in scientific notation. Recognize that it is a small decimal number.

Step 2: Convert to Scientific Notation

Express 0.001 in scientific notation. Since 0.001 is equivalent to 1 divided by 1000, we can write it as \(1 \times 10^{-3}\).

Step 3: Identify the Power

The base is 10, and the exponent is -3. Therefore, 0.001 can be expressed as \(10^{-3}\).

Step 4: Express as a Power

Using the identification from Step 3, write the number 0.001 as a power of 10. The expression is \(10^{-3}\).

Key Concepts

Scientific Notation

Scientific notation is a powerful way to express very large or very small numbers in a concise format. It's especially useful in scientific and mathematical contexts where precision and simplicity are crucial. In scientific notation, a number is written as a product of two factors:

• A decimal number greater than or equal to 1 and less than 10.
• Ten raised to the power of an exponent.

For example, the number 0.001 can seem a bit cumbersome in its decimal form. To convert it to scientific notation, consider its equivalent fraction, 1/1000. This can be expressed as:\[0.001 = 1 \times 10^{-3}\]Here, 1 is the coefficient or significant figure, and \(-3\) is the negative exponent indicating how many places the decimal point has "moved" to the right. This notation makes it much easier to read, compare, and perform calculations with extremely small numbers.

Decimal Conversion

Understanding how to convert numbers between decimal form and scientific notation is a fundamental skill in mathematics. This involves recognizing the order of magnitude a number represents when translated into an exponent of 10.When dealing with small decimal numbers like 0.001, conversion involves shifting the decimal point. For 0.001, you move the decimal three places to the right to reach 1. Each shift corresponds to an exponent increase by -1, resulting in:\[0.001 = 1 \times 10^{-3}\]Learning this conversion tells you how small a number is compared to 1. When understanding decimals in terms of powers of 10, it becomes easier to handle and visualize, especially in fields like science where precision is key.

Negative Exponents

Negative exponents are an essential concept in mathematics that help us deal with fractions and small numbers efficiently. While positive exponents indicate multiplication of a base number, negative exponents show division.For example, a negative exponent like \(10^{-3}\) essentially means the inverse of \(10^3\). It's equivalent to:\[\frac{1}{10^3} = \frac{1}{1000}\]Thus, negative exponents are a shorthand for expressing very small numbers as fractions. This is particularly useful because it simplifies the arithmetic and algebraic handling of tiny values. Whether adding, subtracting, or comparing small quantities, using negative exponents reduces the likelihood of error while keeping calculations neat and manageable.