Problem 10 - Fundamentals of Mathematics

Question

Use the distributive property to compute each value. $$ 80 \cdot 58 $$

Step-by-Step Solution

Answer

The value of \(80 \cdot 58\) is 4640.

Step 1: Break Down the Problem

We want to compute the value of the expression \(80 \cdot 58\) using the distributive property. The distributive property states that \(a(b + c) = ab + ac\). This property can help us simplify large multiplications.

Step 2: Rewrite the Second Number

We will rewrite 58 as a sum that is easy to work with. 58 can be rewritten as \(50 + 8\). This will allow us to distribute 80 over the sum \(50 + 8\).

Step 3: Apply the Distributive Property

Now, let's apply the distributive property: \[80 \cdot (50 + 8) = 80 \cdot 50 + 80 \cdot 8\]

Step 4: Calculate Each Term

Compute each term separately:1. \(80 \cdot 50 = 4000\)2. \(80 \cdot 8 = 640\)

Step 5: Sum the Products

Add the two products we calculated:\[4000 + 640 = 4640\]

Step 6: Verify the Solution

To ensure the calculation is correct, verify by other means such as long multiplication. Confirm \(80 \cdot 58 = 4640\).

Key Concepts

Multiplication

Multiplication is a fundamental arithmetic operation that involves combining groups of equal sizes. It is denoted by the symbol \(\cdot\) or simply by writing numbers next to each other. For instance, computing \(3 \cdot 4\) means having 3 groups of 4, which equals 12. It is a quick and efficient way to add the same number repeatedly.

In our example, we were tasked with multiplying 80 by 58. This might seem cumbersome at first glance, but breaking down numbers can simplify things significantly.

• Understand that multiplication is associative, meaning you can change the grouping of numbers — for example, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• It is also commutative, so the order of numbers does not affect the result — \(a \cdot b = b \cdot a\).
• Knowing these properties helps in re-arranging numbers effectively for easier calculation.

Simplifying Expressions

Simplifying expressions involves reducing them to more manageable forms without changing their values. In arithmetic, this often means breaking down large, unwieldy numbers into parts that are simpler to handle.

In the exercise with 80 and 58, our aim was to find simpler component numbers for 58, which can be split into 50 and 8. A breakdown like this keeps calculations neat and straightforward. Here are some points to remember when simplifying expressions:

• Look for numbers that sum up to or break down a number naturally. For instance, use numbers like 10, 50, or 100 that simplify multiplication.
• Always try to rearrange elements within expressions to align with arithmetic operations that you find simpler.
• Check that the values remain equal after simplification, ensuring accuracy.

Application of Distributive Property

The distributive property is an essential algebraic tool that allows us to multiply a single term across terms within parentheses. It enables the multiplication of each term individually and then sums the results. This is formulated as \(a(b + c) = ab + ac\).

In the given example, we used the distributive property by rewriting the multiplication of 80 with 58 as two distinct operations: multiplying 80 by 50 and 80 by 8. These results were then added together, providing the final result.

• Always identify the sum or difference suitable for distribution in a composite number.
• Apply distribution step-by-step to prevent errors. Compute each resulting expression before summing up.
• Verifying the results through long multiplication or a calculator ensures the distribution was applied correctly.

Understanding and applying the distributive property makes solving complex problems much more feasible in both arithmetic and algebra.