Question

Simplify:

(i) (3 +√3)(2+√2)
(ii) (√3 + √2)^2
(ii) (6+√6)(6-√6)
(iv) (√5 - √2)(√5+√2)

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Answer 1

Answer:-

We have to simplify:-

i) (3 + √3)(2 + √2)

Using (a + b)(c + d) = a(c + d) + b(c + d) we get,

⟹ 3(2 + √2) + √3 (2 + √2)

⟹ 3(2) + 3(√2) + (√3)(2) + (√3)(√2)

⟹ 6 + 3√2 + 2√3 + √(3 × 2). [ ∵ √a × √b = √ab ]

⟹ (3 + √3)(2 + √2) = 6 + 3√2 + 2√3 + √6

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ii) (√3 + √2)²

using (a + b)² = a² + b² + 2ab we get,

⟹ (√3)² + (√2)² + 2(√3)(√2)

⟹ 3 + 2 + 2√6 [ ∵ (√a)² = a ]

⟹ (√3 + √2)² = 5 + 2√6

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iii) (6 + √6)(6 - √6)

using (a + b)(a - b) = a² - b² we get,

⟹ (6)² - (√6)²

⟹ 36 - 6

⟹ (6 + √6)(6 - √6) = 30

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iv) (√5 - √2)(√5 + √2)

Again using a² - b² = (a + b)(a - b) we get,

⟹ (√5)² - (√2)²

⟹ 5 - 2

⟹ (√5 - √2)(√5 + √2) = 3

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Some important formulae:-

• (a + b)² = a² + b² + 2ab

• (a - b)² = a² + b² - 2ab

• a² + b² = (a + b)² - 2ab

• a² + b² = (a - b)² + 2ab

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)

• a³ + b³ = (a + b)³ - 3ab(a + b)

• a³ + b³ = (a + b)(a² - ab + b²)

• a³ - b³ = (a - b)³ + 3ab(a - b)

• a³ - b³ = (a - b)(a² + ab + b²)

Answer 2

Question :

Simplify:

(i) (3 +√3)(2+√2)

(ii) (√3 + √2)²

(iii) (6+√6)(6-√6)

(iv) (√5 - √2)(√5+√2)

Solution :

we have

(i) (3 +√3)(2+√2)

= 3×2 +3 ×√2 +√3×2+√3×√2

= 6+3√2 + 2√3 +√6.

(ii) (√3 + √2)²

= (√3)² + (√2)²+2×√3×√2

[ ∵ ( a - b ) ( a + b) = a² - b² ]

= 3+2+2√6= 5 +2√6.

(iii) (6+√6)(6-√6)

= (6)² - (√6)²

[ ∵ ( a - b ) ( a + b) = a² - b² ]

= (36 - 6 ) = 30

(iv) (√5 - √2)(√5+√2)

= (√5)² - (√2)²

[ ∵ ( a - b ) ( a + b) = a² - b² ]

= 5 - 2 = 3.