Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 49-50: Express the number in scientific form. (a) 427,000 (b) \(0.000000098\) (c) \(810,000,000\)

Step-by-Step Solution

Answer

(a) \(4.27 \times 10^5\), (b) \(9.8 \times 10^{-8}\), (c) \(8.1 \times 10^8\).

Step 1: Understanding Scientific Notation

Scientific notation is a method of expressing numbers as a product of a coefficient (between 1 and 10) and a power of ten. This form is useful for expressing very large or very small numbers concisely.

Step 1: Express 427,000 in Scientific Form

To express 427,000 in scientific notation, we first need to identify the coefficient. Move the decimal point in 427,000 so it's after the first non-zero digit (4), which gives us 4.27. Count how many places the decimal point moves: it moves 5 places to the left. Thus, the scientific notation is \(4.27 \times 10^5\).

Step 2: Express 0.000000098 in Scientific Form

To express 0.000000098 in scientific notation, start by moving the decimal right after the first non-zero digit (9). The number becomes 9.8. Count the number of decimal places moved: it moves 8 places to the right. Therefore, the scientific notation is \(9.8 \times 10^{-8}\).

Step 3: Express 810,000,000 in Scientific Form

For 810,000,000, place the decimal after the first non-zero digit (8), resulting in 8.1. Move the decimal point 8 places to the left. Thus, the scientific notation is \(8.1 \times 10^8\).

Key Concepts

Decimal Point

A decimal point is a crucial component in writing numbers in scientific notation. When converting a standard number to scientific notation, the decimal point needs to be placed immediately after the first non-zero digit. This placement is vital because it helps identify the coefficient, a key part of scientific notation. For example:

• In the number 427,000, moving the decimal to right after 4 gives us 4.27.
• For 0.000000098, placing the decimal after 9 makes it 9.8.
• And with 810,000,000, the decimal after 8 yields 8.1.

The movement of the decimal point also determines the power of ten used, which we'll explore next. This ensures that the coefficient is between 1 and 10, keeping the scientific notation neat and standardized.

Power of Ten

The power of ten in scientific notation indicates how many places the decimal point has been moved. This concept is what helps us express very large or small numbers succinctly. When converting a standard number:

• Moving the decimal point to the left results in a positive power of ten.
• Moving it to the right results in a negative power of ten.

For example, when transforming 427,000 to scientific notation:

• The decimal moves 5 places to the left, so we use a power of ten as 5: \(4.27 \times 10^5\).
• In the case of 0.000000098, the decimal moves 8 places to the right, resulting in \(9.8 \times 10^{-8}\).
• For 810,000,000, moving the decimal 8 places to the left gives \(8.1 \times 10^8\).

This method of power of ten adjustment makes it easier to manage and compute astronomically large or exceedingly small numbers.

Coefficient

In scientific notation, the coefficient is the number that lies between 1 and 10, achieved by moving the decimal point. It provides the base value of the number in this notation before multiplying by the power of ten. Selecting the coefficient correctly is crucial because it determines the accuracy and simplicity of the scientific expression.

• For 427,000, the coefficient becomes 4.27 after adjusting the decimal point.
• In 0.000000098, the coefficient turns into 9.8 once the decimal is moved.
• And in the number 810,000,000, by moving the decimal, the coefficient becomes 8.1.

By ensuring that coefficients remain between 1 and 10, scientific notation guarantees a uniform representation of numbers. This standardization allows for easier and more consistent mathematical operations and comparisons between numbers.