Problem 80 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 8\left\\{9\left[b-2 a+6 c(c+4)-4 c^{2}\right]+4 a+b\right\\}-3 b $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $77b - 112a + 144c^2 + 1728c$.

Step 1: Distribute the terms

Distribute the 8 and 9 to the respective terms inside the parentheses. $$ 8\left\\{9[b-2a+6c(c+4)-4c^2]+4a+b\right\\}-3b = 72(b-2a + 6c(c+4) - 4c^2) + 32a + 8b -3b $$

Step 2: Simplify and expand the terms

Simplify and expand within the parentheses: $$ = 72(b - 2a + 6c^2 + 24c - 4c^2) + 32a + 8b - 3b $$

Step 3: Combine like terms

Combine the like terms to simplify the expression: $$ = 72b - 144a + 144c^2 + 1728c - 3b + 32a + 8b $$

Step 4: Final simplification

Combine the terms once more to get the final simplified expression: $$ = (72b + 8b - 3b) + (-144a + 32a) + (144c^2) + (1728c) = 77b - 112a + 144c^2 + 1728c $$ The simplified expression is: $$ 77b - 112a + 144c^2 + 1728c $$

Key Concepts

Simplification

Simplification in algebra involves making an expression more manageable by reducing it to its simplest form. It's like cleaning up or organizing complicated terms to make them easier to understand. The goal is to rewrite the expression using fewer terms. This means removing unnecessary elements and combining terms that can be grouped together.

• Start by identifying similar terms or operations that can be performed.
• Simplification often involves removing redundant parts by performing basic arithmetic.
• Keep an eye out for terms that can logically be combined to trim down the expression.

In our exercise, simplification is often the final step after using various algebraic properties and operations.

Distributive Property

The distributive property is a fundamental rule in algebra. It allows you to multiply a single term by a group of terms inside a parenthesis. Essentially, you "distribute" the multiplication over addition or subtraction within parentheses. The basic idea can be represented as: \[ a(b + c) = ab + ac \]This property ensures that you can break complex expressions into simpler parts, making them manageable.

• Make sure to multiply each term within the parentheses by the term outside.
• This property helps in expanding expressions so you can work with them more easily later.

In our original problem, we used the distributive property by distributing both 8 and 9 across the terms within the brackets.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variables raised to the same power. This process reduces the number of terms and makes the expression tidier.

• Identify terms with the same variable and same exponent.
• Add or subtract their coefficients accordingly.

In the example, terms containing \( b \), \( a \), and \( c \) with the same exponents were combined to simplify the expression efficiently. This reduces the complexity and makes further operations much simpler.

Expression Expansion

Expression expansion in algebra involves breaking down products into sum or difference terms, often through using the distributive property. This is crucial for further simplification and solving equations. By expanding expressions, we're preparing them to be simplified or evaluated further.

• Expanding helps in visualizing all components of an expression.
• Use expansion to transform complex expressions for easier manipulation.

In our exercise, expression expansion occurred in the step where the distributive property was applied, allowing 9 to be spread across multiple terms, thus breaking initial products down into individual terms for clarity.