Problem 17 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate it. $$(4+6) 9$$

Step-by-Step Solution

Answer

The expression \((4+6)9\) simplifies to \(90\) using the Distributive Property.

Step 1: Identify the Distributive Property

The Distributive Property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). We can apply this property to the expression \((4+6)9\).

Step 2: Apply the Distributive Property

Apply the Distributive Property to the expression by distributing 9 to both 4 and 6: \((4+6)9 = 4 \cdot 9 + 6 \cdot 9\).

Step 3: Perform Multiplication

Calculate the products obtained by distributing: \(4 \cdot 9 = 36\) and \(6 \cdot 9 = 54\).

Step 4: Add the Results

Add the results of the multiplication: \(36 + 54 = 90\). This is the final value after applying the Distributive Property.

Key Concepts

Understanding Equivalent Expressions

In math, equivalent expressions are expressions that are different but represent the same value or quantity. They may look different but will result in the same answer when calculated. Using the Distributive Property, we can write an expression that is equivalent to another. This is a valuable skill because it allows you to simplify complex problems and find alternative ways to solve them.
For example, the expression

• \((4+6)9\)

can be expressed equivalently as

• \(4 \cdot 9 + 6 \cdot 9\)

Both forms will give you the number 90 when simplified. By understanding and creating equivalent expressions, you can improve your problem-solving skills and make calculations more manageable.

Exploring the Property of Operations

The Property of Operations involves rules that apply to arithmetic operations such as addition and multiplication. It includes properties like the Commutative, Associative, and Distributive properties.

The Distributive Property is useful in simplifying expressions and is expressed as

• \(a(b + c) = ab + ac\)

This property helps in breaking down complex calculations into more manageable parts. Take the expression

• \((4+6)9\)

By applying the Distributive Property, you simplify the expression to

• \(4 \cdot 9 + 6 \cdot 9\).

Each step uses both multiplication and addition, demonstrating the efficiency of properties in simplifying mathematical operations. Recognizing these properties allows you to develop a deeper understanding of how numbers interact.

Getting Started with Prealgebra

Prealgebra is an essential foundation for understanding algebra and higher levels of math. It focuses on basic mathematical concepts that prepare you for more complex calculations in algebra.

In prealgebra, you learn to manipulate numbers and expressions, preparing you for equations and problem-solving. One of the key concepts you learn in prealgebra is the Distributive Property. This property allows you to rewrite expressions and simplify your calculations. For example, with prealgebra, you encounter expressions like

• \((4+6)9\)

and learn to apply the Distributive Property to simplify them to a form you can easily calculate,

• \(4 \cdot 9 + 6 \cdot 9\).

Prealgebra helps you build confidence in handling numbers and provides the tools needed to tackle algebraic expressions. By mastering prealgebra, you're well on your way to succeeding in more advanced math courses.