Problem 65 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$-5(10.99)$$

Step-by-Step Solution

Answer

-54.95

Step 1: Distribute

To simplify \(-5(10.99)\) using the distributive property, the operation is multiplication of -5 to 10.99.

Step 2: Perform the multiplication

Perform the multiplication of -5 and 10.99. The multiplication is easily performed as \( -5 \times 10.99 = -54.95 \)

Key Concepts

Simplifying Expressions

When it comes to algebra, simplifying expressions is like tidying up a room—it's all about making things neat and manageable. Simplifying can include several actions such as combining like terms, using the distributive property, or performing arithmetic operations. In our exercise, we simplify the expression \( -5(10.99) \) using the distributive property. This involves spreading out, or distributing, the \( -5 \) across the \( 10.99 \), which is essentially a multiplication task. Think of the \( -5 \) as a tag that applies to each part of the \( 10.99 \) inside the parentheses.

Simplification helps us to see the essence of the expression more clearly, much like decluttering a room helps you focus on individual items better. By simplifying, we can often make other algebraic operations easier to perform or even reveal patterns that weren't initially visible.

Mental Math Multiplication

Multiplication doesn't always require a calculator or paper and pencil. With mental math, you can easily multiply numbers in your head, a handy skill for quick calculations in daily life or while solving algebra problems. To multiply \( -5 \) by \( 10.99 \) mentally, break the problem down. Start by multiplying \( -5 \) by \( 10 \) to get \( -50 \), then multiply \( -5 \) by \( 0.99 \) to get \( -4.95 \) and finally sum these two products to reach \( -54.95 \).

Mental math relies on breaking down complex problems into simpler parts or using known multiplications (like times tables) to simplify the process. It's a skill that improves with practice and can be applied to various mathematical operations.

Algebraic Operations

Algebraic operations are the building blocks for manipulating and solving algebraic expressions and equations. These include addition, subtraction, multiplication, and division, along with more complex actions like factoring, exponentiation, and working with radicals. The operation we used in our textbook problem is multiplication, which is fundamental to the distributive property. This property states that \( a(b + c) = ab + ac \) for any numbers \( a \), \( b \), and \( c \). In our scenario, only multiplication is involved, but if we had a more complex expression, other operations might come into play as well.

It's essential to understand and apply these operations correctly because they serve as the tools to simplify expressions, solve equations, and understand the relationship between variables in algebra.