Question

show that that root 3 + root 5 the whole square is an irrational number​

Answer 1

Show that that root 3 + root 5 the whole square is an irrational number​.

Given:

• Let us assume to the contrary that $($√$3+$√$5)^2$ ia a rational number,
• Then there exists  a and b co-prime integers.

Calculation:

$($√$3+$√$5)^2$$=a/b$

$3+5+2$√$15=a/b$

$8+2$√$15=a/b$

$2$√$15$=$(a/b)-8$

$2$√$15$=$(a-8b)/b$

√$15=(a-8b)/2b$

$(a-8b)/2b$ is a rational number.

• Then √$15$ is also a rational number.
• But as we know √$15$ is an irrational number.
• This is a contradiction.
• This contradiction has arisen as our assumption is wrong.
• Hence  $($√$3+$√$5)^2$ is an irrational number.