Problem 2 - Prealgebra

Question

Express each number in standard form. $$1.4 \times 10^{2}$$

Step-by-Step Solution

Answer

140

Step 1: Understand Scientific Notation

The given number is in scientific notation, where a number is expressed as the product of a number between 1 and 10 and a power of 10. In this case, the number is \(1.4 \times 10^{2}\).

Step 2: Interpret the Exponent

The exponent 2 in \(10^{2}\) indicates that the decimal point of 1.4 should be moved two places to the right, as it is a positive exponent.

Step 3: Move the Decimal Point

Shift the decimal point in 1.4 two places to the right. Moving one place gives 14, and moving another place gives 140.

Step 4: Write the Standard Form

After moving the decimal, the number in standard form is 140.

Key Concepts

Standard Form

Standard form is a common way of writing numbers that allows for simplicity and uniformity. It is especially useful for very large or very small numbers. In standard form, a number is represented as a product of two factors:

• A decimal number, commonly referred to as the coefficient, which ranges from 1 to 10.
• A power of 10, which specifies how many places the decimal point moves.

A positive power indicates a large number, and a negative power suggests a small number. For example, the number 140, expressed in standard form, is given as \(1.4 \times 10^{2}\). This formatting keeps the numbers neat and manageable, making calculations easier.

Exponents

Exponents play a vital role in expressing numbers in standard form and scientific notation. An exponent represents how many times a number, known as the base, is used as a factor. In the expression \(10^{2}\), the base is 10, and the exponent 2 means 10 is multiplied by itself. The exponent's value tells us the number of times we multiply the base 10. If the exponent is positive, the number grows larger, as you multiply by 10 repeatedly, moving the decimal point to the right. Conversely, a negative exponent results in a smaller number, as it indicates division by 10, pulling the decimal point to the left. Understanding exponents helps in accurately interpreting scientific notation and converting it into standard form.

Decimal Point Movement

In scientific notation, moving the decimal point is a crucial step to expressing numbers in standard form. Once you've identified the exponent, you determine how far the decimal needs to shift. For positive exponents, as in \(10^2\), the process involves moving the decimal point to the right.Starting with the number 1.4 in scientific notation \(1.4 \times 10^2\), moving the decimal two places to the right transforms it into 140. The exponent 2 directs you to make this movement, ensuring the accurate conversion to standard form.Decimal point movement is key to bridging the gap between scientific notation and standard form, allowing you to present numbers clearly and understandably. By mastering this skill, you can easily tackle a wide array of numerical expressions with confidence.