Problem 55 - Prealgebra

Question

ALGEBRA Use the Distributive Property to rewrite each expression as an equivalent algebraic expression. (Lesson \(3-1\) ) $$4(2+8)$$

Step-by-Step Solution

Answer

The equivalent expression is 40.

Step 1: Identify the Expression

The expression given is \(4(2+8)\). We need to use the Distributive Property to rewrite it.

Step 2: Apply the Distributive Property

The Distributive Property states that \(a(b + c) = ab + ac\). Here, \(a = 4\), \(b = 2\), and \(c = 8\). Apply the property to get: \(4 \times 2 + 4 \times 8\).

Step 3: Simplify the Expression

Now, perform the multiplication for each term: \(4 \times 2 = 8\) and \(4 \times 8 = 32\). Therefore, the expression becomes \(8 + 32\).

Step 4: Final Simplification

Add the numbers: \(8 + 32 = 40\). The equivalent expression is \(40\).

Key Concepts

Algebraic expressions

Algebraic expressions are like puzzles made of numbers, variables, and operations. They are the foundation of algebra. You can think of them as phrases made up of terms. A term could be a number (which we also call a constant), a single variable (like \(x\) or \(y\)), or a combination of both with coefficients (like \(4x\) or \(3y\)).

These expressions allow you to create mathematical models of real-world situations. When you look at an expression like \(4(2+8)\), each number or variable has a role in the overall picture. Understanding how to manipulate these expressions is key to solving algebraic problems.

• Numbers and Variables: Numbers are fixed values, while variables stand for unknown values that we can change.
• Operations: Addition, subtraction, multiplication, and division are the basic processes used to manipulate expressions.

By learning algebraic expressions, you gain the ability to think logically and solve problems creatively.

Equivalent expressions

Equivalent expressions might look different at first glance but actually represent the same value. This is an important concept in algebra because it helps confirm the solutions to problems or understand different ways to express the same relationship.

For example, when you use the Distributive Property on \(4(2+8)\), you end up with \(8 + 32\), which simplifies to \(40\). Even though \(4(2+8)\) and \(40\) look distinct, they are equivalent expressions because they represent the same value when simplified.

• True Value: Equivalent expressions equal the same numerical value, even if they appear different at first.
• Proof of Understanding: Being able to find equivalent expressions shows a good grasp of algebraic operations and concepts like the Distributive Property.

Spotting equivalent expressions helps in checking your work and ensuring that your solutions make sense.

Simplification in algebra

Simplification in algebra involves making expressions as simple as possible. Simplifying an expression means getting it down to its most basic form where it is easiest to understand or work with. This might involve combining like terms, using the Distributive Property, or performing arithmetic operations.

Take the expression \(4(2+8)\) as an example. You start by using the distributive property to expand it to \(4 \times 2 + 4 \times 8\), then calculate each part individually, leading to the expression \(8 + 32\). Finally, you add the numbers together to get \(40\). Each step makes the expression less complex and easier to handle.

• Less Complex: Simplification reduces the number of terms and makes the expression easier to understand.
• Improved Clarity: A simplified expression shows the true nature of the problem or situation it represents, making it easier to communicate and solve.

Simplified expressions are vital for efficient problem solving, as they remove unnecessary details.