Problem 12 - Fundamentals of Mathematics

Question

Write \(1 x\) in a simpler way.

Step-by-Step Solution

Answer

The expression \(1x\) simplifies to \(x\).

Step 1: Understanding the Exercise

We need to simplify the expression \(1x\). This involves looking for ways to reduce or rewrite the expression using algebraic principles.

Step 2: Applying the Identity Property of Multiplication

The identity property of multiplication states that any number multiplied by 1 remains unchanged. Therefore, when you multiply \(x\) by 1, the result is \(x\) itself. Mathematically, this can be written as: \(1 \times x = x\).

Step 3: Simplifying the Expression

Based on the identity property, we can rewrite \(1x\) in its simplest form. This means dropping the 1, as it does not change the value of \(x\). Thus, the simplified expression is \(x\).

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions involves rewriting them in a more concise form without changing their value. When you simplify, you make the expression easier to work with or understand. For example, in the expression \(1x\), the idea is to express it in the simplest form, getting rid of any unnecessary parts.

Key reasons to simplify expressions include:

• Ease of calculation
• Clarity of interpretation
• Simplification of equations for further manipulation

In our case, we used algebraic principles to simplify \(1x\) to \(x\), showing that these two expressions are exactly the same in value.

Algebraic Principles

Algebraic principles are rules and concepts that guide how algebraic operations are performed. These principles help identify which parts of an expression can be transformed or simplified.

Core algebraic principles include:

• Commutative Property: The order of addition or multiplication does not change the result, such as \(a + b = b + a\) or \(ab = ba\).
• Associative Property: Grouping of numbers does not matter in addition or multiplication: \((a + b) + c = a + (b + c)\).
• Distributive Property: Multiplying a single term by terms inside a parenthesis, like \(a(b + c) = ab + ac\).
• Identity Property: Includes both addition and multiplication; for multiplication, multiplying by 1 does not change the number, as seen in our example \(1 \times x = x\).

In simplifying \(1x\), we specifically relied on the identity property, an essential rule within algebra.

Identity Element in Mathematics

The identity element in mathematics refers to a number or element that, when combined with another number in a given operation, leaves the other number unchanged. In the context of multiplication, the identity element is 1.

Some key points about the identity element include:

• For addition, the identity element is 0: \(a + 0 = a\).
• For multiplication, the identity element is 1: \(a \times 1 = a\).

This concept is crucial because it simplifies expressions and computations. Understanding the role of the identity element helps clarify that multiplying any variable or number by 1 results in the unchanged original variable or number, just as we saw with \(1x = x\). Its utility is not only in operations but also in recognizing when an expression is fully simplified.