Problem 19 - Algebra 1

Question

DECIMAL FORM Rewrite in decimal form. $$ 8.6521 \times 10^{3} $$

Step-by-Step Solution

Answer

Therefore, \(8.6521 \times 10^{3}\) in decimal form is \(8652.1\).

Step 1: Identify the exponent

The number is in scientific notation form \(a \times 10^{n}\). Here, \(n = 3\). This tells us to move the decimal point 3 places to the right.

Step 2: Move the decimal point

Move the decimal point in \(8.6521\) three places to the right. If there aren't enough numbers after decimal, just supplement with zeros. Hence, \(8.6521 \times 10^{3}\) in decimal form becomes \(8652.1\).

Key Concepts

Decimal Form

The decimal form of a number is a way of expressing the number using digits and a decimal point. When writing numbers in decimal form, we aim to present them in a more standardized numerical system, particularly when they arise from computations involving scientific notation.

To convert from scientific notation to decimal form, the goal is to move the decimal point based on the exponent provided, making the number more accessible and easier to read. In the given exercise, we start with the number 8.6521 in scientific notation, and we need to reposition the decimal point according to the exponent to convert it into decimal form.

Scientific notation often involves powers of ten, which determines the direction and number of places you move the decimal point. Understanding the process helps in simplifying calculations and ensuring accurate mathematical expressions in everyday scenarios.

Multiplying by Powers of Ten

Multiplying a number by a power of ten is a straightforward process once you understand how it affects the decimal point. Each power of ten represents how many places the decimal point moves in the number.

• If you multiply by positive powers of ten, the decimal point moves to the right.
• If you multiply by negative powers of ten, the decimal point moves to the left.

In the exercise provided, we multiply by \(10^{3}\), which means the decimal point moves three places to the right. This operation transforms the number from scientific to a more traditional decimal form. It’s a fundamental concept in dealing with various scientific and mathematical calculations because moving the decimal point quickly alters the magnitude of the number.

Exponents

Exponents represent repeated multiplication of a number by itself. In scientific notation, the exponent indicates how many times the base number 10 is used as a factor.

In the problem:

• The number is presented as \(10^{3}\).
• The exponent 3 tells us to multiply the base, 10, by itself three times: \(10 \times 10 \times 10 = 1000\).

Understanding exponents is vital as they are not just limited to scientific notation; they appear in algebraic expressions, physics equations, and real-world applications. They provide a simple way to express very large or very small numbers and perform calculations efficiently. Exponents form the basis of innumerable mathematical models and are integral in developing a greater comprehension of numbers.