Question

Identify the property that justifies each step about the answer and the area below. Line 1: (10x)(5y) line 2: 10*(x*5)*y line 3: 10*(5*x)*y line 4: (10*5)(x*y) line 5: 50xy

Answer 1

Let's break down the given steps and identify the property that justifies each transformation:

1. **Line 1: \( (10x)(5y) \)**
- This is the original expression.

2. **Line 2: \( 10*(x*5)*y \)**
- **Justification: Associative Property of Multiplication.** This property states that the way in which numbers are grouped in multiplication does not change their product. Here, the grouping of \( x \) and \( 5 \) is changed.

3. **Line 3: \( 10*(5*x)*y \)**
- **Justification: Commutative Property of Multiplication.** This property allows the order of factors in multiplication to be changed. In this step, \( 5 \) and \( x \) are reordered.

4. **Line 4: \( (10*5)(x*y) \)**
- **Justification: Associative Property of Multiplication.** Again, we are regrouping the multiplication, but now, instead of rearranging only the first two terms, we are grouping \( 10 \) and \( 5 \) together and keeping \( x \) and \( y \) together.

5. **Line 5: \( 50xy \)**
- **Justification: Simplification of Multiplication (or the definition of multiplication).** Here, we simply multiply \( 10 \) and \( 5 \) together to get \( 50 \) and write it in the standard variable form \( 50xy \).

In summary, the relevant properties used in the transformation of the expression are:
- **Associative Property of Multiplication**
- **Commutative Property of Multiplication**
- **Simplification of Multiplication**