Question

Represent the algebraic expression $$ ((a-b) * c)+7) *((d+4) / x) $$ by a tree, and determine the height of this tree.

Short answer

The tree representation of the given algebraic expression can be visualized as: ``` * / \ * / / \ / \ a - b d+4 x ``` The height of this tree is 3, as there are three edges along the longest path from the root to a leaf node.

Step-by-step solution

Identify the main operations and separators

The given expression is: \[ ((a - b) \cdot c) + 7) \cdot ((d + 4) / x) \] First, we need to identify the main operations and separators - these are the operand signs that split the expression into smaller parts. For this expression, the main operations are multiplication and division, and the main separator is the parenthesis.

Break the expression into its primary components

Next, we will break the expression into its primary components. This will give us a rough idea of the tree's structure. 1. \((a - b) \cdot c\) 2. \((d + 4) / x\) 3. The whole expression is the product of components 1 and 2.

Create the nodes for each primary component

For each primary component, we will create a node in the tree. 1. The node for component 1 will have the operation \(\cdot\), with the left child \((a - b)\) and the right child \(c\). 2. The node for component 2 will have the operation \(/\), with the left child \((d + 4)\) and the right child \(x\).

Break down the child expressions

Now we need to break down the child expressions further. 1. \((a - b)\): This is a simple subtraction operation. Create a node with operation \(-\) and left child \(a\), and right child \(b\). 2. \((d + 4)\): This is a simple addition operation. Create a node with operation \(+\) and left child \(d\), and right child \(4\).

Build the tree

Now that we have all nodes, we can build the tree structure: 1. Create a node with operation \(\cdot\), with left child from step 3.1 and right child from step 3.2. 2. Replace the left child of step 3.1 with the node created in step 4.1. 3. Replace the left child of step 3.2 with the node created in step 4.2. Now we have the tree representation of the given algebraic expression.

Determine the height of the tree

The height of a tree is the length of the longest path from the root node to a leaf node. Here, the longest path is from the main multiplication node, through the nodes representing subtraction and addition, and finally to a leaf node. The tree can be visualized as: ``` * / \ * / / \ / \ a - b d+4 x ``` In this tree, the height is 3, as there are three edges along the longest path from the root to a leaf node.

Key concepts

Algebraic Expressions

Algebraic expressions are fundamental components in mathematics. They are composed of numbers, variables, and operators like addition and multiplication. In the given expression \(((a - b) \cdot c) + 7) \cdot ((d + 4) / x)\), we can identify several key elements:

• **Variables**: These are symbols that represent arbitrary numbers, such as \(a, b, c, d,\) and \(x\).
• **Constants**: These are fixed values, like the number \(7\).
• **Operators**: Symbols that perform mathematical operations: \(-, +, \cdot, /\).

Algebraic expressions allow us to model real-world situations through symbolic representation.
As students progress in mathematics, understanding how to manipulate these expressions becomes crucial for solving more complex problems. A deep grasp of algebraic expressions empowers learners to translate everyday language into mathematical language. This forms the foundation for advanced study, transforming how problems are approached and solved.

Tree Structures in Mathematics

Tree structures are hierarchically arranged representations that help visualize complex mathematical expressions. Imagine a family tree, where members are connected in a clear hierarchy from grandparents to children. Similarly, mathematics uses trees to show relationships between operations and operands.
In the expression given:
\(((a - b) \cdot c) + 7) \cdot ((d + 4) / x)\), we utilize a tree to break down the operations:

• The root node is the main operation, here a multiplication \(\cdot\).
• Each child node represents sub-expressions or operations, like additions or divisions \((+)\) and \((/)\).
• Finally, leaf nodes are the simplest parts of the expression - either variables or numbers \(a, b, etc.\).

This tree structure simplifies understanding how compound expressions are evaluated from the simplest operations up through more complex combinations. Tree structures become particularly useful in computing and programming to parse and execute mathematical expressions efficiently.

Mathematics Problem Solving

Problem-solving in mathematics often requires breaking down complex tasks into simpler steps. For instance, when dealing with algebraic expressions such as our example, the first step is recognizing main operations and their precedence due to operators like \((\cdot)\) or \((/\)).
Consider these strategies:

• **Step-by-step breakdown**: Identify each operation level, from simple to complex, allowing for structured progression.
• **Visual aids**: Drawing helps, such as constructing trees for the expression to see operations' hierarchy.
• **Correct sequencing**: Follow order of operations rules, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In our original exercise, by visualizing the expression through a tree, the problem becomes more manageable, helping identify key steps required to derive the solution. Ultimately, practicing these problem-solving techniques fosters improved proficiency in mathematics, making challenging problems seem approachable and solvable.