Question

a) @(a @ a) ?\( (a) 0 (b) \)a^{4}\( (c) either (a) or (b) (d) neither (a) nor (b) # What is the value of \)(a \\# a) @(a @ a) ?\( (a) 0 (b) \)a^{4}$ (c) either (a) or (b) (d) neither (a) nor (b)

Short answer

Answer: (d) neither (a) nor (b)

Step-by-step solution

Find \(a \\# a\)

In this step, let's find \(a \\# a\). We will use the given definition of this operation: \((a \\# a) = (a^2 – a^3)\). \(a \\# a = a^2 - a^3\)

Find \(a @ a\)

Now, let's find \(a @ a\) by using the given definition of this operation: \((a @ a) = (a^3 – a^2)\). \(a @ a = a^3 - a^2\)

Compute \((a \\# a) @(a @ a)\)

Now that we found \(a \\# a\) and \(a @ a\), let's compute their product: \((a \\# a) @(a @ a) = (a^2 - a^3)(a^3 - a^2)\)

Simplify the expression

Now, let's simplify the expression using the distributive property: \((a^2 - a^3)(a^3 - a^2) = a^5 - a^4 - a^6 + a^4\) Notice that \(-a^4 + a^4\) cancels each other, so we are left with: \(= a^5 - a^6\) Hence, the correct answer is: (d) neither (a) nor (b)

Key concepts

Distributive Property

The distributive property is an essential foundational concept in algebra that helps simplify complex multiplication expressions. Simply put, this property lets you eliminate parentheses by distributing the multiplication over addition or subtraction within the brackets. For example, if you have an expression in the form \( a(b + c) \), applying the distributive property would result in \( ab + ac \).

In this context, we want to apply the distributive property to the expression \( (a^2 - a^3)(a^3 - a^2) \). By doing so, we ensure every term within the first parentheses is multiplied by every term in the second parentheses.

• Multiply \( a^2 \) with \( a^3 \) and \( -a^2 \).
• Multiply \( -a^3 \) with \( a^3 \) and \( -a^2 \).

Then simplify: \( a^2 \cdot a^3 = a^5 \), \( a^2 \cdot -a^2 = -a^4 \), \( -a^3 \cdot a^3 = -a^6 \), and \( -a^3 \cdot -a^2 = a^5 \). After calculating, combine like terms to simplify further.

Polynomial Simplification

Polynomial simplification is the process of reducing expressions to their simplest form. It involves combining like terms and performing mathematical operations like addition, subtraction, multiplication, and sometimes division. This ensures the expression is as straightforward as possible, making it easier to work with further.

In our example, consider the expression \( a^5 - a^4 - a^6 + a^4 \). Here, \( -a^4 \) and \( a^4 \) are like terms.

• Combine these terms to get \( 0 \), effectively removing them from the equation.
• Now, you are left with \( a^5 - a^6 \).

This reduction makes the expression much simpler to understand and manage, leading to the conclusion that the expression evaluates neither to 0 nor \( a^4 \), but to \( a^5 - a^6 \).

Mathematical Expressions

Mathematical expressions are combinations of numbers, operators (like addition and multiplication), and variables. These expressions represent numeric values and relationships. Understanding these expressions fully is critical in algebra, as they allow you to model and solve real-world problems.

In this exercise, the mathematical expressions involve special operations (\( # \) and \( @ \)) with specific definitions:

• The expression \( a \# a \) is defined as \( a^2 - a^3 \).
• The expression \( a @ a \) is defined as \( a^3 - a^2 \).

These definitions form algebraic expressions that are then used in further operations. When these expressions are combined (\( (a \# a) @(a @ a) \)), you create a new expression to simplify using principles like the distributive property and polynomial simplification.