Question

Evaluate (√2 + √3)² (√5 - √2)²

Answer 1

35 -10√10 + 14√6 - 8√15

Explanation:

(√2 + √3)² · (√5 - √2)²

use (a ± b)² = a² ± 2ab + b²

= [(√2)² + 2(√2)(√3) + (√3)²] · [(√5)² - 2(√5)(√2) + (√2)²]

= (2 + 2√6 + 3) · (5 - 2√10 + 2)

= (5 + 2√6)(7 - 2√10)

use FOIL: (a + b)(c + d) = ac + ad + bc + bd

= (5)(7) + (5)(-2√10) + (2√6)(7) + (2√6)(-2√10)

= 35 - 10√10 + 14√6 - 4√60

= 35 - 10√10 + 14√6 - 4√(4 · 15)

= 35 - 10√10 + 14√6 - 4 · √4 · √15

= 35 -10√10 + 14√6 - 4 · 2 · √15

= 35 -10√10 + 14√6 - 8√15

Answer 2

`\left(\sqrt 2 +\sqrt 3 \right)^2 \left( \sqrt 5 - \sqrt 2\right)^2\\\\=\left( 2+2\sqrt 6 +3\right)\left( 5-2√(10)+2\right)\\\\=\left( 5+2\sqrt6\right)\left( 7-2√(10)\right)\\\\=35-10√(10)+14\sqrt6-4√(60)\\\\=35-10√(10)+14\sqrt6-4√(4 * 15)\\\\=35-10√(10)+14\sqrt6-8√(15)\\`