Problem 1 - Prealgebra

Question

Express each number in standard form. $$3.08 \times 10^{-4}$$

Step-by-Step Solution

Answer

The standard form of the number is 0.000308.

Step 1: Understanding Standard Form

A number is in standard form when it is expressed as a product of a coefficient and a power of 10, where the coefficient is at least 1 but less than 10, and the exponent of 10 is an integer. In this case, we need to verify if the expression \(3.08 \times 10^{-4}\) meets these criteria.

Step 2: Verifying the Coefficient

Check if the number 3.08 is between 1 and 10. Since 3.08 is already a number between 1 and 10, it satisfies the condition for the coefficient in standard form.

Step 3: Understanding the Exponent

The exponent on the 10 is \(-4\). In standard form, this means that the decimal must be moved 4 places to the left to represent the number appropriately.

Step 4: Rewriting the Expression in Expanded Form

To express \(3.08 \times 10^{-4}\) in standard form, write the number in full by applying the negative exponent. Move the decimal point in 3.08 four places to the left: 0.000308.

Step 5: Checking Completion

Ensure that the number 0.000308 is indeed the result of applying the exponent to the coefficient. As we have moved the decimal according to the exponent, the conversion to standard form is complete.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more concise form. It's commonly used in science and engineering to simplify numbers, making them easier to read and work with. The format involves writing a number as a product of a coefficient and a power of 10. This means you write the number in the form:

• Coefficient: A number at least 1 but less than 10
• Powers of 10: Multiplied by 10 raised to an exponent

For example, in scientific notation, 3,080 can be written as \(3.08 \times 10^3\). This shows the efficiency of scientific notation, especially for handling numbers that are cumbersome to write in their full form.

Powers of 10

The concept of powers of 10 is fundamental in mathematics, especially when working with scientific notation. Powers of 10 refer to the number of times 10 is multiplied by itself. The exponent denotes this number of times. For example:

• \(10^1 = 10\)
• \(10^2 = 100\)
• \(10^3 = 1000\)

When using scientific notation, the power of 10 tells us how many times we should move the decimal point. If the exponent is positive, the decimal moves to the right indicating a large number. On the other hand, if the exponent is negative, the decimal moves to the left, representing a small number. Negative exponents will be discussed later, but they illustrate just how adaptable powers of 10 can be, making mathematical expressions precise yet uncomplicated.

Coefficients

In scientific notation, the coefficient plays a crucial role in simplifying complex numbers. A coefficient is a number at least 1 but less than 10. This means it provides a standard by which any number can be scaled into scientific notation. For instance, when expressing 3.08 in scientific notation, 3.08 itself is the coefficient. It conforms to the rule as it lies between 1 and 10. The beauty of coefficients is their simplicity. Regardless of how large or small the total number is, the coefficient remains between 1 and 10. This principle keeps scientific notation consistent and manageable across various applications.

Negative Exponents

Negative exponents are used to express very small numbers in scientific notation. They are important for understanding numbers less than one. A negative exponent means we are dealing with a fraction, essentially moving the decimal point to the left.Here's how it works:

• \(10^{-1} = \frac{1}{10} = 0.1\)
• \(10^{-2} = \frac{1}{100} = 0.01\)
• \(10^{-3} = \frac{1}{1000} = 0.001\)

In the problem \(3.08 \times 10^{-4}\), the \(-4\) tells us to move the decimal four places to the left in the number 3.08. This adjustment gives us a much smaller number, 0.000308. Negative exponents are a clever way to simplify and express small quantities easily and accurately.