Selecione a expressão equivalente.

$$(\dfrac{1}{2}\text{x} + \dfrac{1}{3}\text{y})^2 = \text{?}$$
Escolha 1 resposta:
Escolha 1 resposta:
(Escolha A)

$$\dfrac{1}{4}\text{x}^2 + \dfrac{1}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
A

$$\dfrac{1}{4}\text{x}^2 + \dfrac{1}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
(Escolha B)

$$\dfrac{1}{4}\text{x}^2 + \dfrac{2}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
B

$$\dfrac{1}{4}\text{x}^2 + \dfrac{2}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
(Escolha C)

$$\dfrac{1}{4}\text{x}^2 + \dfrac{4}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
C

$$\dfrac{1}{4}\text{x}^2 + \dfrac{4}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$

Question

Selecione a expressão equivalente.
 
 
$$(\dfrac{1}{2}\text{x} + \dfrac{1}{3}\text{y})^2 = \text{?}$$
Escolha 1 resposta:
Escolha 1 resposta:
(Escolha A)   
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{1}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
A
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{1}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
(Escolha B)   
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{2}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
B
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{2}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
(Escolha C)   
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{4}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$
C
 
 
 
$$\dfrac{1}{4}\text{x}^2 + \dfrac{4}{3}\text{x}\text{y} + \dfrac{1}{9}\text{y}^2$$

Accepted Answer

Para resolver essa expressão, precisamos expandir o quadrado do binômio $(\frac{1}{2}x + \frac{1}{3}y)^2$.

A fórmula do quadrado do binômio $(a + b)^2$ é:

$$
(a + b)^2 = a^2 + 2ab + b^2
$$

Substituindo $a = \frac{1}{2}x$ e $b = \frac{1}{3}y$, obtemos:

1. $a^2 = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$
2. $b^2 = \left(\frac{1}{3}y\right)^2 = \frac{1}{9}y^2$
3. $2ab = 2 \times \frac{1}{2}x \times \frac{1}{3}y = \frac{1}{3}xy$

Somando tudo, temos:

$$
\frac{1}{4}x^2 + \frac{1}{3}xy + \frac{1}{9}y^2
$$

**Resposta: A**

A expressão correta é $\frac{1}{4}x^2 + \frac{1}{3}xy + \frac{1}{9}y^2$.

Selecione a expressão equivalente.

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