Problem 54 - Elementary Algebra

Question

For the following problems, use the distributive property to expand the quantities. $$m(u+a)$$

Step-by-Step Solution

Answer

Question: Expand the expression using the distributive property: \(m(u+a)\) Answer: \(mu + ma\)

Step 1: Identify the terms

In our expression, \(m(u+a)\), \(m\) is the term outside the parentheses, and \(u\) and \(a\) are the terms inside the parentheses.

Step 2: Apply the distributive property

Now, we will multiply \(m\) with both \(u\) and \(a\). According to the distributive property, the expression becomes: $$m(u+a) = mu + ma$$

Step 3: Write the final result

After applying the distributive property, the expanded expression is: $$m(u+a) = mu + ma$$

Key Concepts

Algebraic Expressions

Algebraic expressions are like phrases in mathematics. They contain numbers, variables, and operations but lack an equal sign unless part of an equation. Variables in these expressions stand for unknown values and can be represented by letters such as \(x\), \(y\), or \(m\). In our exercise, \(m\), \(u\), and \(a\) are variables.

Understanding algebraic expressions is crucial as they form the basis for many mathematical concepts. These expressions can be simple, involving just one operation, or quite complex, involving multiple layers of operations and terms. In any algebraic expression, terms are the pieces separated by addition or subtraction, like \(mu\) and \(ma\) in the expanded expression of our exercise.

When dealing with expressions, always remember to consider the order of mathematical operations which ensures the expressions are simplified correctly.

Expansion

Expansion in mathematics involves spreading out an expression to show all its parts clearly. When we apply the distributive property, we are expanding an expression. This makes it easier to simplify or evaluate. In the exercise \(m(u+a)\), we use expansion to distribute \(m\) across each term within the parentheses.

Here are some key points about expansion:

• It simplifies expressions into a sum of terms, like transforming \( (u+a) \) into \( u + a \).
• Ensures each component within the parentheses is multiplied by the factor outside, leading to expressions such as \( mu + ma \).
• Helps in recognizing like terms that can be combined, although in our example, \( mu \) and \( ma \) are not combinable.

Remember, expansion is a strategic way to handle expressions, particularly when solving equations or dealing with polynomial expressions.

Mathematical Operations

Mathematical operations are the fundamental processes that we perform on numbers and variables. They include addition, subtraction, multiplication, and division. In the context of the distributive property, multiplication plays a key role.

In our example, we have the operation of multiplication with \(m\) being multiplied by each term inside the parentheses \((u + a)\). Thus, you perform:

• \(m \times u\), resulting in \(mu\)
• \(m \times a\), resulting in \(ma\)

These operations not only include the multiplication itself but also require adding the resulting terms to achieve the expanded expression \(mu + ma\). Understanding how to perform these operations in the correct sequence is essential for correctly applying the distributive property and expanding expressions. This becomes particularly handy when dealing with more complex algebraic expressions.