Problem 283 - Prealgebra

Question

In the following exercises, multiply. $$(104)(256)$$

Step-by-Step Solution

Answer

26624

Step 1: Set Up the Equation

Write down the numbers you need to multiply: 104 and 256.

Step 2: Break Down Numbers

Break 104 and 256 into simpler parts: 104 = 100 + 4 256 = 250 + 6

Step 3: Apply Distributive Property

Use the distributive property to multiply each part: (100 + 4)(250 + 6) = (100 * 250) + (100 * 6) + (4 * 250) + (4 * 6)

Step 4: Simplify Each Multiplication

Calculate each multiplication: (100 * 250) = 25000 (100 * 6) = 600(4 * 250) = 1000(4 * 6) = 24

Step 5: Sum All Parts

Add all the products together: 25000 + 600 + 1000 + 24 = 26624

Key Concepts

Distributive Property

In prealgebra, the distributive property is a fundamental concept used to simplify multiplication. It states that for any numbers a, b, and c, the equation \( a(b + c) = ab + ac \) holds true.
This property allows you to break complex multiplication into smaller, more manageable pieces.
For example, in the given exercise \( (104)(256) \), we can apply the distributive property to break down the numbers and multiply them piece by piece, eventually adding all the parts together to get the final result.
This method not only simplifies the calculation but also helps in understanding the underlying principles of multiplication.

Breaking Down Numbers

Breaking down numbers is a useful technique in prealgebra, especially when dealing with large numbers.
By decomposing numbers into simpler parts, we can make multiplication easier and more intuitive.
In our example, we break down 104 into 100 and 4, and 256 into 250 and 6.
This step prepares us to apply the distributive property effectively.

• 104 becomes 100 + 4
• 256 becomes 250 + 6

Once broken down, we can handle smaller, simpler multiplications before combining all the results, leading to the final answer.

Step-by-Step Multiplication

Using a step-by-step approach to multiplication helps in organizing your work and reducing errors.
Follow these steps for an easier calculation:

• Step 1: Write down the numbers to be multiplied.
• Step 2: Break down the numbers into simpler parts.
• Step 3: Apply the distributive property to set up the multiplication for each part.
• Step 4: Calculate each simplified multiplication separately.
• Step 5: Sum all the products to find the final result.

In the exercise, this method allowed us to handle smaller multiplication tasks: \( (100 * 250) = 25000 \), \( (100 * 6) = 600 \), \( (4 * 250) = 1000 \), and \( (4 * 6) = 24 \). Summing these parts gives us the final answer of 26624.

Prealgebra Problems

Prealgebra problems often involve techniques like breaking down numbers and using the distributive property for multiplication.
These techniques lay the foundation for more advanced mathematics.
They teach students how to approach problems methodically and reduce complex problems into simpler steps.
When working on prealgebra exercises, always remember to:

• Carefully set up each problem.
• Break down large numbers systematically.
• Use properties like the distributive property to simplify calculations.
• Work step by step to ensure accuracy.

By mastering these skills, you'll build a strong foundation for tackling more challenging algebraic concepts in the future.