Problem 10 - Beginning and Intermediate Algebra

Question

Multiply. $$ -6 \cdot 0 $$

Step-by-Step Solution

Answer

0

Step 1: Identify the Rules of Multiplication

Understand that the multiplication of any number with 0 yields 0. This is known as the zero property of multiplication.

Step 2: Apply the Zero Property

Apply the zero property to the given numbers: \(-6 \cdot 0 = 0\). Since multiplying any real number by 0 gives 0, \(-6 \cdot 0\) equals 0.

Key Concepts

Multiplication Rules

Multiplication is an arithmetic operation that combines two or more numbers into a single product. The rules of multiplication help ensure consistent and predictable outcomes. Here are a few important rules to remember:

• Commutative Property: The order of numbers does not affect the product, meaning \(a \cdot b = b \cdot a\).
• Associative Property: You can regroup numbers without changing the result, so \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Distributive Property: You can distribute a number across a sum, for example, \(a \cdot (b + c) = (a \cdot b) + (a \cdot c)\).

Remembering these rules can help with various problems in multiplication, making calculations easier and more straightforward.
Multiplication is often visualized as repeated addition. For example, \(3 \cdot 4\) is the same as adding 3 to itself 4 times. However, knowing and applying these multiplication rules can make solving problems faster and more efficient.

Zero in Multiplication

The zero property of multiplication is a fundamental aspect of this operation. It states that the product of any number and zero is zero.

• No matter how large or small the number is, multiplying it by zero always results in zero.
• For example, \(-6 \cdot 0 = 0\) or \(12345 \cdot 0 = 0\).

This property is essential for simplifying expressions and solving equations in algebra and arithmetic.
Understanding the zero property helps recognize solutions quickly. It is particularly useful in identifying when a particular term or factor in an equation, when multiplied by zero, determines the product to be zero.
The zero property plays a critical role in mathematics and its various applications, such as in solving linear equations where certain terms may effectively "cancel out" due to multiplication by zero.

Real Numbers Multiplication

Real numbers include all the numbers on the number line: positive, negative, whole numbers, fractions, and irrational numbers.

• When multiplying real numbers, the sign of the result is determined by the signs of the numbers being multiplied.
• If both numbers have the same sign, the result is positive. If they have different signs, the result is negative.

For example:

• \(3 \cdot 4 = 12\)
• \(-3 \cdot -4 = 12\) (same signs yield a positive product)
• \(-3 \cdot 4 = -12\) (different signs yield a negative product)

In the case of the zero property, when a real number is multiplied by zero, the result is always zero regardless of the sign. Understanding how multiplication applies to all real numbers, including their interactions with zero, enhances problem-solving skills across a wide range of mathematical topics.