Answer:

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Simplify each of the following expressions:

(i) (3 + √3)(2 + √2) (ii) (3 + √3)(3 - √3) (iii) (√5 + √2)² (iv) (√5 - √2)(√5 + √2)

Solution:

(i) (3 + √3)(2 + √2)

By Distributive property, (a + b) (c + d) = ac + ad + bc + bd

(3 + √3)(2 + √2) = 3 × 2 + 3√2 + √3 × 2 + √3 × √2

= 6 + 3√2 + 2√3 + √6

(ii) (3 + √3)(3 - √3)

Using the identity, (a + b) (a - b) = a² - b²

(3 + √3)(3 - √3) = 3² - (√3)²

= 9 - 3

= 6

(iii) (√5 + √2)²

Using the identity, (a + b) ² = a² + 2ab + b²

(√5 + √2)² = (√5)² + (2×√5×√2) + (√2)²

= (5 + 2√10 + 2)

= 7 + 2√10

(iv) (√5 - √2)( √5 + √2)

Using the identity (a + b) (a - b) = a² - b²

(√5 - √2)( √5 + √2) = (√5)² - (√2)²

= 5 - 2

= 3

Answer:

(i) 7 + 2√10 (ii) 6 + 3√2 + 2√3 + √6 (iii) 5 - 2√6