Question

simplify 1 divided by 1 +root2 + 1 divided by root 2 + root 3 +1 divided by root 3 + root 4

Answer 1

Answer:

Given: \frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}2+326+6+362−6+283

Consider,

\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}2+326+6+362−6+283

Rationalize the denominator of each term,

=\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}\times\frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}\times\frac{\sqrt{6}-\sqrt{3}}{\sqrt{6}-\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}\times\frac{\sqrt{6}-\sqrt{2}}{\sqrt{6}-\sqrt{2}}=2+326×2−32−3+6+362×6−36−3−6+283×6−26−2

Use identity in Denominator a² - b² = ( a + b ) ( a - b )

=\frac{4\sqrt{3}-6\sqrt{2}}{2-3}+\frac{12\sqrt{3}-6\sqrt{6}}{6-3}-\frac{24\sqrt{2}-8\sqrt{6}}{6-2}=2−343−62+6−3123−66−6−2242