Problem 90 - Algebra A Combined Function

Question

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(x+x\) b. \(x \cdot x\) c. \(-x-x\) d. \((-x)(-x)\)

Step-by-Step Solution

Answer

(a) 2x, (b) x^2, (c) -2x, (d) x^2.

Step 1: Additive Identity

To simplify the expression \(x + x\), we recognize that this is the same as adding the variable \(x\) to itself. According to the properties of addition, \(x + x\) is equivalent to \(2x\). Therefore, the simplified form of \(x + x\) is \(2x\).

Step 2: Multiplicative Identity

To simplify the expression \(x \cdot x\), we apply the property of exponents, which states that multiplying variables with the same base results in the base raised to the power of the sum of the exponents. Since \(x\) can be rewritten as \(x^1\), \(x \cdot x\) becomes \(x^2\). Thus, the simplified form is \(x^2\).

Step 3: Subtracting Integers

For \(-x - x\), consider how subtraction and negative signs work. Subtracting \(x\) from \(-x\) is the same as adding the negative of \(x\) to \(-x\), i.e., \(-x - x\) = \(-(x + x)\). Therefore, \(-x - x = -2x\).

Step 4: Negative Multiplication

For \((-x)(-x)\), apply the rule that the product of two negative numbers is positive. Thus, \((-x)(-x) = x^2\), because multiplying the negative signs results in a positive sign, and multiplying \(x\) by itself gives \(x^2\).

Key Concepts

Additive Identity

In algebra, simplification often involves understanding fundamental properties. One such property is the additive identity. The additive identity property states that the sum of zero and any number is that number itself.
For example, if we have a number, such as 5, adding zero to it gives us 5 again. This seems fairly straightforward, but it's a crucial concept.
In a given expression like \(x + x\), we're not using zero directly. Instead, we're using the concept that adding a number to itself is essentially doubling it. Thus, \(x + x\) becomes \(2x\). This is because we have taken the variable and added it to itself, in line with the concept of repetitive addition.
Though it seems simple, the additive identity underpins how we approach addition, even with variables.

Multiplicative Identity

The multiplicative identity focuses on the number 1. According to this property, multiplying any number by 1 gives you the number itself. For example, \(5 \times 1 = 5\).
When dealing with algebraic expressions, this property helps us manipulate equations without changing their values. For example, with \(x \cdot x\), we apply the properties of exponents.
We know that multiplying a number by itself can be rewritten using exponents: in this case, \(x \cdot x = x^2\).
Here, each \(x\) can be considered as \(x^1\), and when we multiply like bases, we add their exponents, resulting in \(x^{1+1} = x^2\).

• This property becomes crucial when simplifying and solving equations.
• It also helps in factoring expressions and manipulating complex algebraic forms.

Understanding the multiplicative identity is essential for working with exponents and more complex algebraic operations.

Negative Number Multiplication

Multiplying negatives might initially seem counterintuitive, but there's a simple rule: multiplying two negative numbers results in a positive number.
This rule is based on the way negative numbers work in pair mathematics. Consider \((-1)\cdot(-1)\). By definition, this returns a positive result, 1. Similarly, \((-x)(-x)\) becomes \(x^2\).
In the expression \((-x)(-x)\), you flip the negative signs, turning the result positive as multiplication removes the negative aspect of each factor.
Here are some quick points:

• Negative \(\times\) Negative = Positive
• Negative \(\times\) Positive = Negative
• Positive \(\times\) Negative = Negative
• Positive \(\times\) Positive = Positive

Remembering these rules helps in dealing with complex algebra problems that involve negatives.

Properties of Exponents

Exponents simplify repeated multiplication. Understanding the fundamental properties of exponents helps us in various mathematical operations.
Consider the problem of \(x \cdot x = x^2\). This uses the basics of exponents, where you multiply the same base and add their exponents. For instance, \(x^m \cdot x^n = x^{m+n}\).
Other properties include:

• \(x^0 = 1\), assuming \(x eq 0\)
• \((x^m)^n = x^{m\cdot n}\)
• \(x^{-n} = \frac{1}{x^n}\) for \(x eq 0\)

Mastering these principles will enable you to tackle equations efficiently and handle the specifics of algebraic manipulation or simplifications.