Question

Simplify each of the following expressions:
1)(3+V3)(2+V2)
2)(3+V3)(3-V3)
3)(V5+V2) sq
4)(V5-V2)(V5+V2)

Answer 1

Answer:

1) 6 + 3√2 + 2√3 + √6

2) 6

3)7 + 2√10

4) 3

Step-by-step explanation:

1) (3 + √3)(2 + √2)

→ 3(2 + √2) + √3(2 + √2)

→ 3(2) + 3(√2) + √3(2) + √3(√2)

→ 6 + 3√2 + 2√3 + √6

2) (3 + √3)(3 - √3)

Identity : (a + b)(a - b) = a² - b²

Here, a = 3, b = √3

→ (3)² - (√3)²

→ 9 - 3

→ 6

3) (√5 + √2)²

Identity : (a + b)² = a² + b² + 2ab

Here, a = √5, b = √2

→ (√5)² + (√2)² + 2(√5)(√2)

→ 5 + 2 + 2√10

→ 7 + 2√10

4) (√5 - √2)(√5 + √2)

Identity : (a + b)(a - b) = a² - b²

Here, a = √5, b = √2

→ (√5)² - (√2)²

→ 5 - 2

→ 3

Other identities which are frequently used :

• (a - b)² = a² + b² - 2ab

• (a + b)² - (a - b)² = 4ab

Answer 2

Answer:

Question:-

1) (3 + √3)(2 + √2)

2)(3 + √3)(3 - √3)

3) (√5 + √2)^2

4) (√5 - √2)(√5 + √2)

$\rule{200}{1}$

Solution:-

$\sf 1)(3 + \sqrt{3}) + (2 + \sqrt{2})\\\\\Rightarrow\sf 3(2 + \sqrt{2}) + \sqrt{3}(2 + \sqrt{2}) \\\\\Rightarrow\large{\boxed{\sf\green{6 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6} }}}$

$\rule{100}{2}$

$\sf 2) (3 + \sqrt{3}) (3 - \sqrt{3}) \\\\\Rightarrow\sf (3)^2 - (\sqrt{3})^2 \\\\\Rightarrow\large{\boxed{\sf\pink{6}}}$

$\rule{100}{2}$

$\sf 3) (\sqrt{5} + \sqrt{2})^2 \\\\\Rightarrow\sf (\sqrt{5})^2 + (\sqrt{2})^2 + 2\sqrt{10} \\\\\Rightarrow\sf 5 + 2 + 2\sqrt{10} \\\\\Rightarrow\large{\boxed{\sf\red{7 + 2\sqrt{10} }}}$

$\rule{100}{2}$

$\sf 4) (\sqrt{5} - \sqrt{2}) (\sqrt{5} + \sqrt{2}) \\\\\Rightarrow\sf (\sqrt{5})^2 - (\sqrt{2})^2 \\\\\Rightarrow\sf 5 - 2 \\\\\Rightarrow\large{\boxed{\sf\blue{3}}}$

$\rule{200}{2}$

Some more identities:-

• $\rm a^2 - b^2 = (a + b)(a - b)$
• $\rm (a + b)^2 = a^2 + b^2 + 2ab$
• $\rm (a - b)^2 = a^2 - b^2 + 2ab$