Problem 63 - Elementary Algebra

Question

Write the following problems using scientific notation. $$ 0.376 $$

Step-by-Step Solution

Answer

Question: Rewrite the decimal number 0.376 in scientific notation. Answer: 3.76 × 10^(-1)

Step 1: Identify the coefficient and exponent

First, we need to identify a coefficient between 1 and 10 and an exponent that, when multiplied together, would result in the original decimal number 0.376. In this case, the coefficient will be 3.76, and the exponent will be -1 since we need to move the decimal point one place to the right in order to obtain 3.76 from the original number 0.376.

Step 2: Write the number in scientific notation

Now that we have the coefficient and exponent, we can write the number in scientific notation. The scientific notation will be: $$ 3.76 \times 10^{-1} $$

Key Concepts

Exponent

In scientific notation, the exponent plays a crucial role. It shows how many times the base number, typically 10, is multiplied by itself.

When the exponent is positive, it indicates how many places the decimal point moves to the right. This results in a larger number. However, in our case with a decimal like 0.376, the exponent is negative. A negative exponent means you move the decimal point to the left, which results in a smaller number.

For example, in the scientific notation for 0.376, which is written as \(3.76 \times 10^{-1}\), the exponent -1 indicates that the decimal point from 3.76 must move leftward one spot to return to 0.376.

• The exponent helps simplify numbers by reducing the number of zeros needed.
• Negative exponents are common when describing very small numbers, as they help represent the number in a more digestible form.
• The base in scientific notation is usually 10 because we operate on a decimal system.

Decimal Numbers

Decimal numbers represent a fraction of a whole number and are expressed using a decimal point. They are important in scientific notation for simplifying calculations.

Every digit after the decimal point shows a fraction of ten. For example, in the number 0.376:

• The '3' is in the tenths place.
• The '7' is in the hundredths place.
• The '6' is in the thousandths place.

The further a digit is to the right of the decimal point, the smaller its value.

When converting decimal numbers to scientific notation, the goal is to adjust the decimal place so that the resulting number, also known as the "coefficient," falls between 1 and 10. By reducing the number of decimal places, scientific notation helps handle very large or very small numbers more easily.
Creating a coefficient from a decimal number involves moving the decimal point to form a number between 1 and 10, like converting 0.376 to 3.76.

Coefficient

The coefficient in scientific notation is a value between 1 and 10 that is derived from the original number. It represents the significant figures or digits of a number.

In our example with the number 0.376, the coefficient is adjusted to be 3.76 for scientific notation. Here's how the process works:

• We shift the decimal point of 0.376 one place to the right to get 3.76.
• As a result, the exponent decreases by one, leading to an exponent of -1, to balance the equation back to the original value.

A coefficient achieves clarity by presenting the number in a standard format. This enables easy multiplication with the stated power of 10 (as indicated by the exponent).

One critical aspect of finding the correct coefficient is to ensure it accurately reflects the original number's magnitude.

This means it maintains the same quantity of significant figures as the original number.