Problem 30 - Elementary and Intermediate Algebra

Question

Multiply. \(0 \cdot(-4.7)\)

Step-by-Step Solution

Answer

0

Step 1: Understand the multiplication of zero

When any number is multiplied by zero, the product is always zero. This is a fundamental property of multiplication.

Step 2: Apply the property

Here, we have the expression \(0 \cdot (-4.7)\). Based on the property from Step 1, regardless of what the other number is (in this case, -4.7), the result of multiplying it by zero will be zero.

Key Concepts

headline of the respective core concept

Multiplying by zero might seem straightforward, but it's a crucial concept in mathematics. When you multiply any number by zero, the result is always zero. This is because zero represents having none of something. For instance, if you have zero groups of 5 apples, you end up with zero apples. This applies universally in math.
In the exercise, our expression is \(0 \cdot (-4.7)\). Here, you can see that zero is multiplying \(-4.7\). Based on this principle, without even performing any complex multiplication, we know the result is zero.
This concept applies no matter how large or complicated the other number is. Whether it’s 0 multiplied by 10,000 or by \(-4.7\), the result remains the same.

headline of the respective core concept

Fundamental multiplication properties help us understand how numbers interact. One of these properties is the commutative property, which states that changing the order of numbers in a multiplication operation does not change the result. For example, \(3 \cdot 4 = 4 \cdot 3\). Another is the associative property, which indicates that how we group numbers in multiplication does not affect the outcome: \((2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4)\).
Multiplying by zero falls under the distributive property, which fundamentally changes the operation. If you multiply zero with any number, you spread zero across it, effectively nullifying that number.
This exercise \(0 \cdot (-4.7)\) is a straightforward application of distributive property. Even though \(-4.7\) is negative, zero multiplied by any number results in zero, simplifying our calculations significantly.

headline of the respective core concept

The zero property of multiplication is a key idea in understanding how numbers relate. It states that any number multiplied by zero will equal zero. It does not matter if the number is positive, negative, a fraction, or a large integer.
Let’s look in detail: Imagine you have a bag of zero candies. If you distribute those candies among five friends, each friend gets zero candies. The distribution process, or multiplication here, confirms that zero multiplied by any distribution factor results in zero.
So, in our exercise, we have \(0 \cdot (-4.7)\). Here, zero is multiplied by a negative decimal. Regardless of this added complexity, the property's rule prevails and the result again is zero. This property simplifies many mathematical problems, ensuring solutions are easier to find in diverse contexts.