Problem 67 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 5(9+8) $$

Step-by-Step Solution

Answer

5(9+8) = 85

Step 1 - Understand the Distributive Property

The distributive property states that for any numbers a, b, and c: \[a(b + c) = ab + ac\]

Step 2 - Identify the Values

In the given expression, the values are:\[a = 5\], \[b = 9\], and \[c = 8\]

Step 3 - Apply the Distributive Property

Using the distributive property, distribute the 5 to both 9 and 8:\[5(9 + 8) = 5 \times 9 + 5 \times 8\]

Step 4 - Perform the Multiplications

Calculate each term separately:\[5 \times 9 = 45\] and \[5 \times 8 = 40\]

Step 5 - Combine the Results

Add the results from the multiplications:\[45 + 40 = 85\]

Key Concepts

algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. An example of an algebraic expression is \( 3x + 4y - 7 \). Unlike equations, they do not have an equal sign. These expressions can represent various situations in real life, like calculating costs, measuring areas, and more. Understanding how to manipulate and simplify them is key to algebra. An important part of this manipulation is learning how to apply the distributive property, which helps in breaking down expressions into simpler parts.

distribution in mathematics

Distribution in mathematics refers to the distributive property of multiplication over addition. This property allows you to multiply a single term by each term inside a parenthesis individually: \[a(b + c) = ab + ac\]. For example, in the expression \(5(9 + 8)\), the number 5 must be distributed to both 9 and 8. This means you'll multiply 5 by 9 and also 5 by 8, then add the results. This property is particularly useful in algebra when simplifying expressions or solving equations. By practicing distribution, you get better at recognizing patterns and simplifying complex problems step by step.

basic multiplication

Basic multiplication is one of the fundamental operations in arithmetic. It involves combining equal groups of objects, and it can be thought of as repeated addition. For instance, \(3 \times 4\) is the same as adding 3 four times: \(3 + 3 + 3 + 3 = 12\). In algebra, multiplication is often used to simplify expressions and solve equations. Understanding multiplication is also crucial when applying the distributive property. Taking the example \(5(9 + 8)\), we first multiply 5 by 9 to get 45 and then 5 by 8 to get 40. Adding these products together gives us 85. Mastering basic multiplication helps in handling more complex mathematical tasks with ease.