Find the Product / Simplify

These problems involve expanding algebraic expressions, typically using the difference of squares identity: $(a + b)(a – b) = a^2 – b^2$.

1. $(3x + 5y)(3x – 5y)$

2. $(7a – 2b)(7a + 2b)$

3. $(p + \frac{4}{q})(p – \frac{4}{q})$

4. $(2m + \frac{1}{n})(2m – \frac{1}{n})$

5. $(9x + 11)(9x – 11)$

Evaluate Numerical Expressions

These problems require using algebraic identities to simplify mental multiplication.

6. $104 \times 96$

7. $498 \times 502$

8. $1003 \times 997$

Find the Value

These problems focus on calculating the difference between two squares.

9. $61^2 – 39^2$

10. $125^2 – 75^2$

Using the identity for the difference of squares,

, here is the explanation and solution for each problem from the document:

Section 1: Finding the Product

1. $(3x + 5y)(3x – 5y)$

   • Explanation: Here, $a = 3x$ and $b = 5y$. Square both terms and subtract.

   • Result: $(3x)^2 – (5y)^2 = \mathbf{9x^2 – 25y^2}$

2. $(7a – 2b)(7a + 2b)$

   • Explanation: Here, $a = 7a$ and $b = 2b$. The order of the plus/minus signs doesn’t change the identity.

   • Result: $(7a)^2 – (2b)^2 = \mathbf{49a^2 – 4b^2}$

3. $(p + \frac{4}{q})(p – \frac{4}{q})$

   • Explanation: Even with fractions, the rule stays the same. $a = p$ and $b = \frac{4}{q}$.

   • Result: $p^2 – (\frac{4}{q})^2 = \mathbf{p^2 – \frac{16}{q^2}}$

4. $(2m + \frac{1}{n})(2m – \frac{1}{n})$

   • Explanation: Square the first term $(2m)$ and the second term $(\frac{1}{n})$.

   • Result: $4m^2 – \frac{1}{n^2}$

Section 2: Numerical Evaluation

To solve these without a calculator, we rewrite the numbers as $(a + b)$ and $(a – b)$ based on a round number.

5. $104 \times 96$

   • Explanation: Rewrite as $(100 + 4)(100 – 4)$.

   • Result: $100^2 – 4^2 = 10,000 – 16 = \mathbf{9,984}$

6. $498 \times 502$

   • Explanation: Rewrite as $(500 – 2)(500 + 2)$.

   • Result: $500^2 – 2^2 = 250,000 – 4 = \mathbf{249,996}$

7. $1003 \times 997$

   • Explanation: Rewrite as $(1000 + 3)(1000 – 3)$.

   • Result: $1000^2 – 3^2 = 1,000,000 – 9 = \mathbf{999,991}$

Section 3: Differences of Squares

8. $61^2 – 39^2$

   • Explanation: Use the identity in reverse: $a^2 – b^2 = (a + b)(a – b)$.

   • Step: $(61 + 39)(61 – 39) = 100 \times 22$

   • Result: $\mathbf{2,200}$

9. $125^2 – 75^2$

   • Explanation: Apply the same reverse identity.

   • Step: $(125 + 75)(125 – 75) = 200 \times 50$

   • Result: $\mathbf{10,000}$

Section 4: Simplification

10. $(9x + 11)(9x – 11)$

    • Explanation: Standard difference of squares where $a = 9x$ and $b = 11$.

    • Result: $(9x)^2 – 11^2 = \mathbf{81x^2 – 121}$