Problem 34 - Elementary Algebra Within Reach

Question

$$ \text { In Exercises 33-46, simplify the expression. } $$ $$ -7(5 a) $$

Step-by-Step Solution

Answer

-35\(a\)

Step 1: Apply the relevant algebraic rules

Use properties of exponents, radicals, or algebraic identities to simplify the expression.

Step 2: State the simplified result

The simplified expression is -35\(a\).

Key Concepts

Multiplication of Integers

Multiplying integers is a fundamental concept in mathematics that involves combining two whole numbers to yield a product. The process is straightforward:

• If one or both of the numbers are negative, the product will also be negative.
• If both integers are positive, or both are negative, the result will be positive.

To multiply integers like \(-7 \times 5\), identify the signs first:

• \(-7\) is negative
• 5 is positive

Then, use the rule: different signs result in a negative product. Multiply the absolute values (ignore the signs momentarily): \(7 \times 5 = 35\). Since the initial numbers had different signs, the product is negative, resulting in \(-35\).
Understanding this basic principle aids in tackling more complex mathematical problems.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables (like \(a\), \(b\), or \(x\)), and operations (addition, subtraction, multiplication, and division). These expressions allow us to generalize mathematical concepts and find solutions for variable scenarios.
In the expression '-35\(a\)', we have:

• -35 is the coefficient, a numerical factor multiplying the variable.
• \(a\) is the variable, representing a value that can change.

When simplifying algebraic expressions, it often involves performing operations to combine like terms or reduce expressions to a simpler form.
Although '-35\(a\)' cannot be simplified further without additional information about \(a\), the goal is to maintain a clear structure that easily communicates the expression's components.

Integer Operations

Integer operations encompass addition, subtraction, multiplication, and division. These operations are the building blocks for more advanced concepts in mathematics.

• **Addition/Subtraction**: Combine or subtract integers by considering their signs; same signs add, different signs subtract.
• **Multiplication/Division**: As covered earlier, multiply or divide considering the rules for signs to determine whether the outcome is positive or negative.

It is essential to practice these operations to ensure competence in handling both straightforward and complex mathematical problems.
For example, working through \(-7 \times 5\) develops the skill to manage integer operations involving different signs.
These operations are crucial not only for solving algebraic expressions but for understanding the relationships between numbers in various mathematical contexts.