If x, y, and z are three natural numbers, then write the rule that shows that multiplication distributes over addition.

Answer

To show that multiplication distributes over addition for three natural numbers \( x \), \( y \), and \( z \), we can use the distributive property of multiplication. Here’s the step-by-step solution:

### Step 1: State the Distributive Property
The distributive property states that for any numbers \( a \), \( b \), and \( c \):
\[ a \times (b + c) = (a \times b) + (a \times c) \]
In this case, we will replace \( a \) with \( x \), \( b \) with \( y \), and \( c \) with \( z \).

### Step 2: Apply the Property
Using the property, we can write:
\[ x \times (y + z) = (x \times y) + (x \times z) \]

### Step 3: Example Calculation
Let’s take an example where \( x = 2 \), \( y = 3 \), and \( z = 4 \).

1. **Calculate the Left-Hand Side (LHS)**:
\[ LHS = x \times (y + z) = 2 \times (3 + 4) = 2 \times 7 = 14 \]

2. **Calculate the Right-Hand Side (RHS)**:
\[ RHS = (x \times y) + (x \times z) = (2 \times 3) + (2 \times 4) = 6 + 8 = 14 \]

### Step 4: Conclusion
Since both sides are equal:
\[ LHS = RHS = 14 \]
This confirms that multiplication distributes over addition.

### Final Statement
Thus, we have shown that:
\[ x \times (y + z) = (x \times y) + (x \times z) \]
for natural numbers \( x \), \( y \), and \( z \).

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