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

Question

Simplify the following expressions:

(i) (\(\sqrt{3}\) + \(\sqrt7\))²

(ii) (\(\sqrt5\) - \(\sqrt{3}\))²

(iii) (2\(\sqrt5\) + 3\(\sqrt2\))²

Answer 1

(i)  (\(\sqrt{3}\) + \(\sqrt7\))² = (\(\sqrt{3}\))² + 2 x \(\sqrt{3}\) x \(\sqrt7\) + (\(\sqrt7\))²

Because: (a + b)² = (a)² + 2 x \(\sqrt{a}\) x \(\sqrt{b}\) + (b)²

= 3 + 2\(\sqrt{3\times 7}\) + 7

= 10 + 2\(\sqrt21\)

(ii)  (\(\sqrt5\) - \(\sqrt{3}\))² = (\(\sqrt5\))² - 2 x \(\sqrt5\) x \(\sqrt3\) + (\(\sqrt3\))²

(a)² - 2 x \(\sqrt{a}\) x \(\sqrt{b}\) + (b)²

= 5 - 2\(\sqrt{5\times 3}\) + 3

= 8 - 2\(\sqrt15\)

(iii)  (2\(\sqrt5\) + 3\(\sqrt2\))² = (2\(\sqrt5\))² + 2(2\(\sqrt5\)) x (3\(\sqrt2\)) + (3\(\sqrt2\))²

= 2² x \(\sqrt5\)² + 2 x 2 x 3 x \(\sqrt{5\times 2}\) + 3² x \(\sqrt2\)²

= 4 x 5 + 12\(\sqrt{5\times 2}\) + 9 x 2

= 20 + 12\(\sqrt10\) + 18

= 38 + 12\(\sqrt10\)