Problem 62 - Prealgebra

Question

Simplify each expression. $$3(k-10)$$

Step-by-Step Solution

Answer

The simplified expression is \(3k - 30\).

Step 1: Distribute the 3

To simplify the expression, you first need to distribute the 3 to both terms inside the parentheses. This means multiplying 3 by each term inside. So, distribute as follows: 3 multiplied by k is 3k and 3 multiplied by -10 is -30.

Step 2: Write the Simplified Expression

After performing the distribution, write down the simplified terms in a single expression. Combine the terms to get the simplified expression:\[3k - 30\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that helps to simplify expressions and equations. It involves spreading a multiplier across terms within parentheses. In mathematical terms, if you have an expression like \( a(b + c) \), the distributive property allows you to rewrite it as \( ab + ac \).
This means you multiply the number outside the parentheses by each term inside. In our original exercise with the expression \( 3(k-10) \), we use the distributive property to simplify it:

• Multiply \(3\) by \(k\) to get \(3k\).
• Then multiply \(3\) by \(-10\) to get \(-30\).

Applying the distributive property is crucial because it transforms expressions into a more workable form, making it easier to solve equations or perform further algebraic manipulations.

Expression Simplification

Expression simplification is a process in mathematics where we transform expressions to their simplest form. This is done to make expressions easier to understand or use in further calculations. Simplifying an expression means reworking it, so it has fewer terms or easier terms while keeping the same value.
In the exercise, after applying the distributive property, the expression \(3(k-10)\) becomes \(3k - 30\).
Here are some key steps for simplifying expressions:

• Use distributive property to eliminate parentheses.
• Combine like terms, if any. (Though not needed in this example)
• Ensure the expression is in its simplest form.

Simplifying expressions can help in solving equations, comparing expressions, or simply for ease of interpretation, especially in more complex problems.

Mathematics Education

Mathematics education focuses on equipping learners with essential concepts and techniques to engage with numbers and logic effectively. Understanding algebraic principles, such as the distributive property and expression simplification, is core to this education. These skills can be applied not just in pure math but in real-life situations as well.
Here are some reasons why these concepts are important in mathematics education:

• Foundation for Advanced Math: Knowing how to distribute and simplify provides a base for solving equations and inequalities.
• Critical Thinking: These skills develop logical and analytical thinking as students learn to manipulate expressions to reach conclusions.
• Problem Solving: Simplification can turn a difficult problem into a manageable one by breaking it down into simpler parts.

With continuous practice of these principles, students can enhance their mathematical fluency, paving the way for success in more complex math and related fields.