Simplify the following expressions : (i) (4 + √ 7) (3 + √ 2) (ii) (3 + √3) (5 - √ 2) (iii) (√ 5 - 2)(√ 3 - √ 5)

Question

Simplify the following expressions:

(i) (4 + \(\sqrt7\)) (3 + \(\sqrt2\))

(ii) (3 + \(\sqrt3\)) (5 - \(\sqrt2\))

(iii) (\(\sqrt5\) - 2)(\(\sqrt3\) - \(\sqrt5\))

Answer 1

(i) (4 + \(\sqrt7\))(3 + \(\sqrt2\))

= 4 x 3 + 4 x \(\sqrt2\) + \(\sqrt7\) x 3 + \(\sqrt7\) x \(\sqrt2\)

= 12 + 4\(\sqrt2\) + 3\(\sqrt7\) + \(\sqrt14\)

(ii) (3 + \(\sqrt3\))(5 - \(\sqrt2\))

= 3 x 5 + 3 x (-\(\sqrt2\)) + \(\sqrt3\) x 5 + \(\sqrt3\) x (-\(\sqrt2\))

= 15 - 3\(\sqrt2\) + 5\(\sqrt3\) - \(\sqrt3\) x 2

= 15 - 3\(\sqrt2\)  + 5\(\sqrt3\) - \(\sqrt6\)

(iii) (\(\sqrt5\) - 2)(\(\sqrt3\) - \(\sqrt5\))

= \(\sqrt5\) x \(\sqrt3\) + \(\sqrt5\) x (-\(\sqrt5\)) + (-2) x \(\sqrt3\) +(-2) x (-\(\sqrt5\))

= \(\sqrt5\) x 3 - \(\sqrt5\) x 5 - 2\(\sqrt3\) + 2\(\sqrt5\)

= \(\sqrt15\) - \(\sqrt{5^2}\) - 2\(\sqrt3\) + 2\(\sqrt5\)

= \(\sqrt15\) - 5 - 2\(\sqrt3\) + 2\(\sqrt5\)