Problem 22 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (x+8) 7 $$

Step-by-Step Solution

Answer

The expression (x+8) 7 without parentheses is \(7x + 56\).

Step 1: Distribute Multiplication Over Addition

To start with, apply the distributive property on the expression which means to multiply the 7 with both the 'x' and the '8'. This leads to the expression \(7*x + 7*8\)

Step 2: Complete the Multiplication

The next step requires the multiplication operations to be performed. This results in \(7x + 56\).

Key Concepts

Expression Simplification

Simplifying expressions is a crucial skill in algebra that reduces complexity and makes calculations easier. It involves rewriting expressions in a more straightforward form without changing their value. In our example,
when you apply the distributive property to \((x+8)7\), you are simplifying the expression.
This results in an expression without parentheses: \(7x + 56\). This simplified form is easier to work with if further operations are needed.
Simplification can involve several steps, such as combining like terms or performing arithmetic operations.

• Identify expressions that can be expanded, like those with parentheses.
• Perform operations such as addition, subtraction, multiplication, or division as required.
• Rearrange terms to combine like terms efficiently.

Learning to simplify expressions helps in solving equations and performing calculus operations more successfully.

Multiplicative Operation

Multiplicative operations in algebra involve multiplying coefficients with variables or numbers. This is seen in our example where the number 7 is multiplied by both 'x' and 8.
When considering the expression \((x + 8)7\), break it down to \(7*x + 7*8\), which shows two separate multiplicative operations.Here's how it works:

• Multiply each term inside the parentheses by the term outside.
• This results in a product for each term, maintaining the integrity of the expression.
• Each multiplication step follows basic arithmetic rules.

Multiplicative operations help expand and further simplify expressions, especially when using the distributive property.

Algebraic Expression

An algebraic expression combines numbers, variables (like 'x'), and arithmetic operations (like addition and multiplication). Our example expression is \((x+8)7\), which includes both a variable and numbers.
The goal of working with algebraic expressions is to manipulate them to make solving equations easier. Key elements include:

• Variables represent unknown values that can change or be solved for.
• Numbers provide concrete values, aiding in solving the unknowns.
• Operations between variables and numbers define the expression's structure.

Understanding algebraic expressions are foundational in algebra, providing the basis for learning about equations, functions, and calculus.

Combining Like Terms

Combining like terms is a process of simplifying an expression by merging terms with the same variables. In the expression \(7x + 56\), the terms cannot be combined further because they consist of different parts: '7x' is a term with a variable, whereas '56' is merely a constant.Steps involved in combining like terms:

• Identify terms with the same variables and the same exponents.
• Add or subtract coefficients of these similar terms.
• Re-structure the expression with only essential terms.

This process reduces complexity and simplifies the solving of equations. It is a vital skill for mastering algebra and preparing for more advanced math concepts.