Verify the distributive law for rational numbers.
The distributive law for rational numbers states that if p, q, and r are rational numbers, then p(q + r) = pq + pr. We can verify this by taking p = 8/3, q = 1/2, and r = 3/4. First, we evaluate the LHS: p(q + r) = (8/3) × (1/2 + 3/4). To add the fractions inside the brackets, we make the denominators equal: (8/3) × (2/4 + 3/4) = (8/3) × (5/4) = 10/3. Next, we evaluate the RHS: pq + pr = (8//2) + (8//4). Multiplying the terms gives 4/3 + 2. Converting 2 to a fraction with denominator 3 gives 4/3 + 6/3 = 10/3. Since LHS = RHS, the distributive law is verified.
Explanation
The textbook defines the distributive law for rational numbers as p(q+r) = pq + pr. To verify it, a student must substitute rational numbers for p, q, and r, and independently calculate both sides to show they are equal. Using the values from the textbook's example (p=8/3, q=1/2, r=3/4) perfectly demonstrates this property through basic fraction arithmetic.
Solution Steps
Step 1: State the distributive law: p(q + r) = pq + pr. Let p = 8/3, q = 1/2, and r = 3/4.
Step 2: Calculate LHS = p(q + r) = (8/3) × (1/2 + 3/4) = (8/3) × (5/4) = 10/3.
Step 3: Calculate RHS = pq + pr = (8//2) + (8//4) = 4/3 + 2 = 10/3.
Step 4: Compare both sides. Since LHS = RHS (10/3 = 10/3), the distributive law is verified.