Problem 14 - Prealgebra

Question

Express each number in standard form. $$9.01 \times 10^{-2}$$

Step-by-Step Solution

Answer

The standard form of the number is 0.0901.

Step 1: Understand the Standard Form

The standard form of a number is a way of expressing numbers conveniently. It is written as the product of a number between 1 and 10 and a power of 10. In this exercise, the number is already in this form as it is expressed as a product of 9.01 and \(10^{-2}\).

Step 2: Calculate the Product

Calculate the product by multiplying 9.01 with \(10^{-2}\), which is equivalent to moving the decimal point two places to the left. This changes 9.01 to 0.0901.

Step 3: Verify the Result

Ensure that the number obtained is correctly expressed in standard numerical form, which is the more familiar decimal notation that we use in everyday math. Hence, 9.01 × \(10^{-2}\) in decimal form is 0.0901.

Key Concepts

Scientific Notation

Scientific notation is a special way of writing numbers that makes it easier to handle very large or very small quantities. It uses powers of ten to express a number in a more concise format. A number in scientific notation is usually written as the product of a number (called the coefficient) and a power of ten. For example, in "9.01 \times 10^{-2}", 9.01 is the coefficient, and \(10^{-2}\) is the power of ten.

Scientific notation is especially useful in scientific and engineering contexts where precise measurements are common. By using this notation, you can easily compare very different scales, simplifying calculations and saving space on paper.

• The coefficient is always a number greater than or equal to 1 and less than 10.
• The exponent on the power of ten indicates how many places to move the decimal point.

For the given problem, 9.01 \times 10^{-2} is already in scientific notation, indicating that 9.01 needs to be moved two decimal places to the left, resulting in 0.0901 in decimal form.

Decimal Notation

Decimal notation is the standard way of writing numbers that we use in everyday math. It represents numbers as a series of digits with a decimal point to separate the whole number from the fractional part. When a number is converted from scientific notation to decimal notation, you are translating the "power of ten" movement into a tangible position of digits.

In decimal notation, there are no exponents involved, which means the number is shown in its most accessible form. For the number given in the exercise, you convert "9.01 \times 10^{-2}" into decimal notation by moving the decimal point of 9.01 two places to the left. This movement reflects the \(10^{-2}\) power, hence giving you the standard decimal format of 0.0901.

• Decimal notation is very familiar because it is how we generally see numbers displayed.
• It's the practical form for reading, writing, and interpreting numbers daily.

Thus, understanding how to shift between scientific to decimal notation ensures you can comfortably handle both forms without confusion.

Powers of Ten

Powers of ten are a way to scale numbers up or down using the base number ten raised to an exponent. This concept is fundamental in both scientific and decimal notations, as it defines the distance the decimal point needs to be moved.

• A positive exponent, such as \(10^2\), results in moving the decimal point to the right, making the number larger.
• A negative exponent, such as \(10^{-2}\), shifts the decimal point to the left, reducing the number's size.

In the problem "9.01 \times 10^{-2}", the exponent of -2 means that you need to move the decimal point of 9.01 two places left to convert it into decimal notation.

Powers of ten are essential because they provide a clear and consistent way to express relative sizes. Once you understand how these exponents work, converting between scientific and decimal notation becomes straightforward and intuitive, allowing for seamless transitions in various mathematical problems.