Problem 298 - Prealgebra

Question

In the following exercises, simplify. $$86 \times 29$$

Step-by-Step Solution

Answer

The product of 86 and 29 is 2494.

Step 1: Write the Problem

Write the multiplication problem: \(86 \times 29\).

Step 2: Break Apart 29

Use the distributive property to break apart 29. Write it as \(29 = 20 + 9\). So, the problem becomes \(86 \times (20 + 9)\).

Step 3: Distribute 86

Apply the distributive property to multiply 86 by each part of the sum. \[86 \times (20 + 9) = 86 \times 20 + 86 \times 9\]

Step 4: Multiply Each Term

Calculate each term separately. \(86 \times 20 = 1720\) and \(86 \times 9 = 774\).

Step 5: Add the Products

Add the products obtained from the previous step. \(1720 + 774 = 2494\).

Step 6: Write the Final Answer

The simplified form of \(86 \times 29\) is 2494.

Key Concepts

distributive property

The distributive property is a key principle in mathematics, especially useful in algebra and arithmetic. It allows you to simplify complex multiplication problems by breaking them into smaller, more manageable parts. In our example, we used the distributive property to transform the problem from \(86 \times 29\) into a simpler form.

The property states that:

• \(a \times (b+c) = (a \times b) + (a \times c)\)
• This means you can distribute the multiplication across the terms inside the parentheses.

In our problem, we transformed 29 into 20 + 9, then distributed 86 to both parts:
\[86 \times (20 + 9) = 86 \times 20 + 86 \times 9\]
This step is crucial as it makes lengthy multiplication easier by splitting it into simpler parts.

multiplication steps

Multiplication can be daunting if approached all at once, but breaking it into steps can simplify the task. Here's how we can solve \(86 \times 29\) step-by-step:

• Step 1: Identify the problem: \(86 \times 29\).
• Step 2: Break apart 29 into 20 + 9, so the problem becomes \(86 \times (20 + 9)\).
• Step 3: Apply the distributive property: \(86 \times 20 + 86 \times 9\).
• Step 4: Calculate each term separately: \(86 \times 20 = 1720\) and \(86 \times 9 = 774\).
• Step 5: Add the products: \(1720 + 774 = 2494\).

By following these steps, the multiplication problem becomes straightforward and manageable.

simplifying arithmetic expressions

Simplifying arithmetic expressions involves reducing them to their simplest form using mathematical properties and operations. In this case, we simplified \(86 \times 29\) using the distributive property. Here are some steps to simplify arithmetic expressions effectively:

• Identify components: Break down complex numbers into smaller, easier to handle parts.
• Apply properties: Use properties like the distributive property to rewrite expressions.
• Perform operations: Carry out the multiplication or addition as simplified terms.
• Combine results: Add the results of individual calculations to get the final answer.

By consistently applying these steps, you can simplify not only complex multiplication but a wide range of arithmetic expressions.