Problem 60 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 7 a(a-4) $$

Step-by-Step Solution

Answer

Question: Simplify the expression 7a(a - 4). Answer: 7a^2 - 28a

Step 1: Distribute 7a to both terms inside the parentheses

To distribute 7a to both terms inside the parentheses, multiply 7a by a and then by -4. This will result in the following expression: $$ 7a\cdot a + 7a\cdot (-4) $$

Step 2: Perform the multiplications

Now, perform the multiplications: $$ 7a^2 - 28a $$

Step 3: Combine like terms (if any)

In this case, there are no like terms to combine as the two terms are distinct. Thus, the simplified expression is: $$ 7a^2 - 28a $$

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. They do not include an equality sign like equations do. Instead, they represent a value or range of values depending on what the variables stand for. In the expression \(7a(a-4)\), we have a variable, \(a\), which acts like a placeholder for any number. Algebraic expressions allow you to model real-world situations. They are powerful tools in algebra, enabling you to describe patterns and solve problems.

Here's how algebraic expressions are structured:

• Terms: Parts of the expression separated by '+' or '-' signs.
• Coefficients: Numerical parts multiplied by the variables (e.g., 7 in \(7a\)).
• Variables: Symbols, often letters, that stand for unknown values (e.g., \(a\)).

Mastering algebraic expressions is key to understanding and performing operations in algebra.

Multiplication in Algebra

Multiplication in algebra works similarly to multiplication in arithmetic, but here it involves both numbers and variables. When multiplying algebraic expressions, each term in the first expression should be multiplied by every term in the second. This principle is known as distribution.

In the problem \(7a(a-4)\), you need to apply the distributive property, where you multiply \(7a\) by each term in the parenthesis. Here's how:

• Multiply \(7a\) by \(a\)
• Multiply \(7a\) by \(-4\)

The result of these operations is \(7a^2\) and \(-28a\). Combining these you get \(7a^2 - 28a\). Multiplication in algebra often requires careful attention to ensure proper application of the rules.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms with the same variables raised to the same exponents. This makes expressions easier to work with and understand.

When dealing with algebraic expressions, terms are 'like' if they have the same variable factors. For example, in the expression \(3x + 4x\), \(3x\) and \(4x\) are like terms and can be combined to \(7x\).

In the exercise \(7a^2 - 28a\), after performing the multiplication, you would check for like terms. However, here, the terms, \(7a^2\) and \(-28a\), are not like terms because they have different powers of \(a\). Therefore, no further simplification is needed in combining them. Understanding how and when to combine like terms is crucial in simplifying algebraic expressions effectively.