for any whole numbers a, b, and c, a times (b times c) = (a times b) times c.
This is known as the associative property of multiplication. It states that for any whole numbers a, b, and c, the result of multiplying a by the product of b and c is equal to the result of multiplying the product of a and b by c. In other words:
a × (b × c) = (a × b) × c
This property holds true for all whole numbers, irrespective of the specific values chosen for variables a, b, and c.
To prove the statement "for any whole numbers a, b, and c, a times (b times c) = (a times b) times c," we can use the associative property of multiplication.
The associative property states that for any three numbers, a, b, and c, the order of multiplication can be changed without changing the result.
Let's break down the equation step by step:
1. Start with a times (b times c):
a * (b * c)
2. Multiply b by c:
(a * (b * c)) = (a * bc)
3. Now, let's consider the second expression: (a times b) times c:
(a * b) * c
4. Multiply a by b:
((a * b) * c) = (ab * c)
By comparing steps 2 and 4, we can see that both expressions simplify to the same result, which is (a * bc).
Therefore, we have shown that a times (b times c) is equal to (a times b) times c using the associative property of multiplication.
The statement you mentioned is known as the associative property of multiplication. It states that for any whole numbers a, b, and c, the product of a, b, and c will be the same regardless of the grouping.
To understand why this property holds true, let's go through a step-by-step explanation:
1. Start with the left-hand side of the equation: a times (b times c).
- First, multiply b and c together: b times c.
- Then, multiply the result by a: a times (b times c).
2. Now, let's move to the right-hand side of the equation: (a times b) times c.
- First, multiply a and b together: a times b.
- Then, multiply the result by c: (a times b) times c.
To prove that these two sides are equal, we need to show their equivalence.
3. Consider the product a times (b times c):
- This expression means that we multiply a by the result of multiplying b and c.
- So, the multiplication of b and c is performed first, then we multiply the result by a.
- This multiplication order can also be represented using parentheses as: (a times (b times c)).
4. Next, consider the product (a times b) times c:
- This expression means that we multiply a and b together, and then multiply the result by c.
- So, we multiply a by b, and then multiply the result by c.
- This multiplication order can also be represented using parentheses as: ((a times b) times c).
By comparing steps 3 and 4, we can see that both expressions are structured the same way. They have the same order of multiplication operations, just written differently with the use of parentheses.
Hence, we can conclude that a times (b times c) is equal to (a times b) times c, which proves the associative property of multiplication for whole numbers.