Problem 9 - College Algebra with Modeling and Visualization

Question

$$ 5 \times 10^{3}=_______ $$

Step-by-Step Solution

Answer

5000

Step 1: Understand the Problem

The goal is to express the equation \( 5 \times 10^3 \) in its numerical form. This means, we need to calculate the value of the expression \( 5 \text{ multiplied by } 10 ext{ raised to the power of } 3 \).

Step 2: Calculate the Power

Calculate \( 10^3 \). This represents 10 raised to the power of 3, which means multiplying 10 by itself three times: \( 10 \times 10 \times 10 = 1000 \).

Step 3: Perform the Multiplication

Now, multiply 5 by the result of the previous calculation (1000). So, \( 5 \times 1000 = 5000 \).

Step 4: Provide the Final Answer

The equation \( 5 \times 10^3 \) equals 5000. Therefore, the blank is filled with the value 5000.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact and simple form. It is particularly useful in fields like science and engineering, where precision and clarity are important. The basic format of scientific notation involves writing a number as the product of a coefficient and a power of ten. For example, a number like 5000 can be represented as \( 5 \times 10^3 \).
This format consists of two parts:

• The coefficient: This is a number greater than or equal to 1 and less than 10. In our example, it is 5.
• The power of ten: This indicates how many times the coefficient should be multiplied by 10. In the example, the power of ten is 3, which means the coefficient (5) should be multiplied by 1000 (or \( 10^3 \)).

Using scientific notation simplifies arithmetic operations on very large or very small numbers and makes it easier to read and understand the scaled factor of the number.

Arithmetic Operations

Arithmetic operations are the processes that form the foundation of mathematics and involve adding, subtracting, multiplying, or dividing numbers. When dealing with expressions in scientific notation, such as \( 5 \times 10^3 \), multiplication plays a key role.
Let's break it down:

• Power Calculation: Compute the power of ten first. In \( 5 \times 10^3 \), you first calculate \( 10^3 \), which equals 1000. This is similar to multiplying 10 by itself three times (\( 10 \times 10 \times 10 \)).
• Multiplication: After calculating the power of ten, multiply the result by the coefficient. So, in this case: \( 5 \times 1000 \). Perform the simple arithmetic operation to find that it equals 5000.

Understanding these operations ensures accurate results and helps alleviate confusion, especially when managing expressions involving exponents.

Powers of Ten

Powers of ten are a mathematical shorthand that determines how many times ten is multiplied by itself. They are fundamental in understanding scientific notation and simplifying arithmetic operations involving large numbers. A power of ten looks like \( 10^n \), where \( n \) is a whole number.
Here's why they are useful:

• Expression: They allow for expressing extremely large or tiny quantities conveniently. For example, \( 10^3 \) represents 1000, while \( 10^6 \) signifies a million.
• Simplification: Powers of ten help in simplifying calculations, as multiplying or dividing numbers is streamlined. Understanding that \( 10^n \) means adding zeros to 1 helps in quickly determining magnitudes.
• Scalability: They offer a scalable way to handle numbers across different magnitudes without losing precision, which is essential in various scientific calculations and engineering tasks.

Using powers of ten, like in \( 5 \times 10^3 \), clarifies the scale of the number and simplifies the multiplication process by clearly indicating the number's magnitude.