1. $$ (a-3)(a+3) $$ 2. $$ (2 a-7 b)(3 a-2 b) $$ 3. $$ \left(\frac{1}{2} x \frac{1}{3} y\right)\left(\frac{1}{3} x-\frac{1}{2} y\right) $$ c. $$ \left(x-\frac{y}{3}\right)^{2} $$

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UpStudy AI · Accepted Answer

Sure! Let's expand each of the given algebraic expressions step by step.

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### 1. $$ (a - 3)(a + 3) $$

This is a **difference of squares**. The formula for the difference of squares is:
$$
(x - y)(x + y) = x^2 - y^2
$$

Applying this to the given expression:
$$
(a - 3)(a + 3) = a^2 - 3^2 = a^2 - 9
$$

**Expanded Form:**
$$
a^2 - 9
$$

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### 2. $$ (2a - 7b)(3a - 2b) $$

We'll use the **distributive property** (also known as the FOIL method for binomials) to expand this expression.

$$
(2a - 7b)(3a - 2b) = 2a \cdot 3a + 2a \cdot (-2b) + (-7b) \cdot 3a + (-7b) \cdot (-2b)
$$

Calculating each term:
$$
= 6a^2 - 4ab - 21ab + 14b^2
$$

Combine like terms:
$$
= 6a^2 - 25ab + 14b^2
$$

**Expanded Form:**
$$
6a^2 - 25ab + 14b^2
$$

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### 3. $$ \left(\frac{1}{2}x + \frac{1}{3}y\right)\left(\frac{1}{3}x - \frac{1}{2}y\right) $$

Again, we'll use the **distributive property** to expand this expression.

$$
\left(\frac{1}{2}x + \frac{1}{3}y\right)\left(\frac{1}{3}x - \frac{1}{2}y\right) = \frac{1}{2}x \cdot \frac{1}{3}x + \frac{1}{2}x \cdot \left(-\frac{1}{2}y\right) + \frac{1}{3}y \cdot \frac{1}{3}x + \frac{1}{3}y \cdot \left(-\frac{1}{2}y\right)
$$

Calculating each term:
$$
= \frac{1}{6}x^2 - \frac{1}{4}xy + \frac{1}{9}xy - \frac{1}{6}y^2
$$

Combine like terms ($$-\frac{1}{4}xy + \frac{1}{9}xy$$):
$$
= \frac{1}{6}x^2 - \left(\frac{1}{4} - \frac{1}{9}\right)xy - \frac{1}{6}y^2 
$$
$$
= \frac{1}{6}x^2 - \frac{5}{36}xy - \frac{1}{6}y^2
$$

**Expanded Form:**
$$
\frac{1}{6}x^2 - \frac{5}{36}xy - \frac{1}{6}y^2
$$

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### c. $$ \left(x - \frac{y}{3}\right)^2 $$

To expand this, we'll use the formula for the square of a binomial:
$$
(x - y)^2 = x^2 - 2xy + y^2
$$

Applying this to the given expression:
$$
\left(x - \frac{y}{3}\right)^2 = x^2 - 2 \cdot x \cdot \frac{y}{3} + \left(\frac{y}{3}\right)^2
$$

Simplifying each term:
$$
= x^2 - \frac{2}{3}xy + \frac{y^2}{9}
$$

**Expanded Form:**
$$
x^2 - \frac{2}{3}xy + \frac{y^2}{9}
$$

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If you need further assistance with these expressions or have other questions, feel free to ask!
1. $$ (a - 3)(a + 3) = a^2 - 9 $$ 2. $$ (2a - 7b)(3a - 2b) = 6a^2 - 25ab + 14b^2 $$ 3. $$ \left(\frac{1}{2}x + \frac{1}{3}y\right)\left(\frac{1}{3}x - \frac{1}{2}y\right) = \frac{1}{6}x^2 - \frac{5}{36}xy - \frac{1}{6}y^2 $$ c. $$ \left(x - \frac{y}{3}\right)^2 = x^2 - \frac{2}{3}xy + \frac{y^2}{9} $$

1. \( (a-3)(a+3) \) 2. \( (2 a-7 b)(3 a-2 b) \) 3. \( \left(\frac{1}{2} x \frac{1}{3} y\right)\left(\frac{1}{3} x-\frac{1}{2} y\right) \) c. \( \left(x-\frac{y}{3}\right)^{2} \)

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