Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$0(-11)$$

Step-by-Step Solution

Answer

The result of the multiplication of 0 with -11 is 0.

Step 1: Perform the multiplication

As per the multiplication rule, any number multiplied by zero is always zero. Hence, \(0 \times -11 = 0\)

Step 2: Identify the algebraic structure

Determine the type of algebraic problem.

Step 3: Apply algebraic techniques

Use factoring, expanding, or systematic methods.

Step 4: Simplify and solve

Simplify expressions and solve for unknowns.

Step 5: State the result

Write the final answer.

Step 6: Conclude with the answer

The result of the multiplication of 0 with -11 is 0.

Key Concepts

Zero property of multiplication

In multiplication, the zero property is a fundamental principle that states that any number multiplied by zero equals zero. It's as simple as that. This rule makes multiplying by zero one of the easiest operations in math, because no matter how complex or large the numbers you are working with, the moment you multiply them by zero, the result will always be zero. Here are a few things to remember about the zero property of multiplication:

• This property applies to all real numbers, whether they are positive, negative, whole numbers, or fractions.
• The order of multiplication does not affect the result, so either way, the product will be zero, e.g., \(a \times 0 = 0\) or \(0 \times a = 0\).
• It is an essential part of number theory and forms the foundation of more advanced mathematical concepts.

Understanding the zero property helps simplify complex algebraic expressions and solve equations more easily.

Negative numbers

Negative numbers are numbers that are less than zero and are typically used to represent values below a defined point, like temperature below freezing, debt in finance, or a depth below sea level.

• In mathematics, they are usually denoted with a minus sign \(-\) before the number, such as \(-3\) or \(-11\).
• Negative numbers are part of the real number system, which also includes positive numbers and zero.
• When two negative numbers are multiplied, the product is positive, because the negative signs cancel each other out.
• Conversely, multiplying a negative number by a positive number, as seen with \(0 \times -11\), results in a negative product, but here the zero property takes precedence, leading to zero.

Working with negative numbers is crucial in algebra and helps in understanding real-world scenarios that involve losses or decreases.

Algebraic multiplication

Algebraic multiplication is a method used in algebra where expressions are multiplied together. This can involve variables, constants, or both. The basic principles are:

• Each term in one expression is multiplied by each term in the other expression.
• The product of numbers follows the same rules as arithmetic multiplication, including the commutative, associative, and distributive properties.
• Variables are multiplied by applying the laws of exponents, where terms are combined and powers are added, such as \(x^2 \times x^3 = x^{2+3} = x^5\).
• Algebraic multiplication allows us to simplify expressions and solve equations efficiently.

When dealing with complex expressions, it's important to remember these rules to maintain accuracy. Algebraic multiplication is a skill that becomes easier with practice and is key to tackling more advanced math problems.