Problem 44 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$7(2 a+2)+4(5 a-1)$$

Step-by-Step Solution

Answer

The simplified expression is \(34a + 10\).

Step 1: Apply the Distributive Property to the First Term

Begin by applying the distributive property to the expression \(7(2a + 2)\). This means multiplying 7 by each term inside the parentheses. The expression becomes \(7 \times 2a + 7 \times 2 = 14a + 14\).

Step 2: Apply the Distributive Property to the Second Term

Next, apply the distributive property to the expression \(4(5a - 1)\). Multiply 4 by each term inside the parentheses. The expression becomes \(4 \times 5a - 4 \times 1 = 20a - 4\).

Step 3: Combine Like Terms

Now, combine the two expressions from the previous steps: \(14a + 14\) and \(20a - 4\).Add the like terms: constants with constants and terms with \(a\) with terms with \(a\).Combine \(14a\) and \(20a\) to get \(34a\); combine the constants \(14\) and \(-4\) to get \(10\).

Step 4: Write the Simplified Expression

The simplified expression is now formed by the combined terms from Step 3.The complete expression is \(34a + 10\).

Key Concepts

Combining Like Terms

Combining like terms is a vital step in simplifying algebraic expressions. It involves grouping and adding together terms that have identical variables and exponents. When you approach combining like terms, it helps to look for terms that share the same variable and power. In our exercise, after using the distributive property, we receive the terms:
- \(14a\)- \(20a\)- Constants: \(14\) and \(-4\)To combine these, start by focusing on the terms with the same variable, in this case, \(a\). Add \(14a\) and \(20a\) to get \(34a\). Next, sum the constants: \(14 + (-4) = 10\).
Combining like terms effectively transforms complex expressions into simpler, more manageable ones. It is like cleaning up your desk; everything gets ordered, and it becomes easier to see everything clearly!

Simplifying Expressions

Simplifying expressions means to reduce them to their simplest form while keeping the original value intact. It's like putting a jumble of puzzle pieces into a coherent picture.
To simplify the expression from our exercise, we executed several steps:

• Applied the distributive property to break down the problem.
• Combined like terms to clean up the expression.

After combining like terms, we found our simplified expression: \(34a + 10\).
The goal of simplifying is to get the expression to be as efficient and as easy to interpret as possible. This makes it easier to use in further math operations or solve equations.

Prealgebra Concepts

Prealgebra lays the foundational skills needed for understanding and performing algebra. It's like building the base of a house; without it, the structure isn't stable.
In our exercise, prealgebra concepts appear everywhere:

• We see the distributive property when multiplying numbers across sums and differences in parentheses.
• The idea of combining like terms reinforces basic arithmetic skills while introducing essential algebraic principles.

Understanding these basic concepts arms students with the tools to handle more complex operations as they advance.
Prealgebra doesn't just prepare you for the next math class; it equips you with a logical framework that you'll use across math and science disciplines. It's why getting a solid grasp in prealgebra makes future math challenges much easier to conquer.