Problem 62 - Algebra 1: Concepts and Skills

Question

Use the distributive property and mental math to simplify the expression. $$ -9(1.02) $$

Step-by-Step Solution

Answer

-9.18

Step 1: Understand Distributive property

The distributive property is an algebraic rule that states that a multiplied by the sum of b and c is equal to a * b plus a * c. It can be written as a (b + c) = a*b + a*c. In this case -9 is a while 1.02 is b.

Step 2: Apply Distributive Property

Applying the distributive property to -9(1.02), we see that we can distribute -9 to 1.02. -9 * 1.02 = -9.18.

Step 3: Perform The Final Multiplication

Multiplying -9 by 1.02, we use mental math or calculator to multiply. The result of this multiplication is -9.18.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition and multiplication. They form the backbone of algebra and are used to express a wide range of mathematical relationships. In this exercise, we work with a simple algebraic expression that features a multiplication operation: \-9(1.02)\. Here, \-9\ is the coefficient, and \(1.02\) is the term being multiplied.

• **Coefficient**: A number that multiplies a variable or another number.
• **Term**: A part of an expression separated by addition or subtraction.

Understanding how to manipulate and simplify these expressions using properties like the distributive property is a key part of learning algebra. Mastery of algebraic expressions provides the means to solve equations and model real-world problems effectively.

Simplification

Simplification is an important process in mathematics where an expression is reduced to its simplest form. This makes mathematical problems easier to solve and understand. To simplify the expression \-9(1.02)\ using the distributive property, follow these steps:

• Identify the coefficient \-9\ and distribute it across the term \(1.02\).
• Carry out the multiplication: \-9 \, \times \, 1.02 = -9.18\.

The distributive property helps in breaking down complex expressions into manageable parts, facilitating easier computation. In our example, repeating the operation ensures that we simplify accurately.

Mental Math

Mental math is a skill that allows you to perform calculations in your head without the use of paper, pencil, or calculator. It's incredibly useful for quick computations and helps build numerical intuition.When applying mental math to simplify \-9(1.02)\, break it down into simpler components:

• Think of \(1.02\) as \(1 + 0.02\).
• Use the distributive property: Calculate \-9 \, \times \, 1 = -9\ and \-9 \, \times \, 0.02 = -0.18\.

Adding these results gives you \-9 - 0.18 = -9.18\. Practicing mental math encourages flexibility in thinking and develops skills in estimation, which is useful for checking work for errors and solving problems more efficiently.