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AQA GCSE · Question 25 · Algebra
Here are three terms: xy, x², 5y². Alec multiplies two of these terms. Work out the three possible fully simplified answers.
Here are three terms: xy, x², 5y². Alec multiplies two of these terms. Work out the three possible fully simplified answers.
How to approach this question
1. Identify the three possible pairs of terms to multiply.
- Pair 1: (xy) and (x²)
- Pair 2: (xy) and (5y²)
- Pair 3: (x²) and (5y²)
2. Multiply the terms in Pair 1:
(xy) * (x²) = x * y * x * x = x¹ * x² * y = x¹⁺² * y = x³y.
3. Multiply the terms in Pair 2:
(xy) * (5y²) = 5 * x * y * y² = 5 * x * y¹⁺² = 5xy³.
4. Multiply the terms in Pair 3:
(x²) * (5y²) = 5 * x² * y² = 5x²y².
5. List the three simplified answers.
Full Answer
We need to find the product of each possible pair of the three terms: `xy`, `x²`, and `5y²`.
**Possibility 1: `xy` multiplied by `x²`**
(xy) × (x²) = x × y × x²
Group the x terms together: (x × x²) × y
Using the laws of indices (aᵐ × aⁿ = aᵐ⁺ⁿ), we have x¹ × x² = x¹⁺² = x³.
So, the result is **x³y**.
**Possibility 2: `xy` multiplied by `5y²`**
(xy) × (5y²) = x × y × 5 × y²
Group the numbers and y terms: 5 × x × (y × y²)
Using the laws of indices, we have y¹ × y² = y¹⁺² = y³.
So, the result is **5xy³**.
**Possibility 3: `x²` multiplied by `5y²`**
(x²) × (5y²) = x² × 5 × y²
It is conventional to write the number first, then the variables in alphabetical order.
So, the result is **5x²y²**.
The three possible simplified answers are x³y, 5xy³, and 5x²y².
Common mistakes
✗ Incorrectly applying the laws of indices (e.g., multiplying powers instead of adding them, x² * x = x²).
✗ Forgetting the coefficient (the number 5).
✗ Missing one of the three possible pairs.
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