Question

show that
$( \sqrt{3} + \sqrt{5} ) {}^{2}$
is an irrational number
​

Answer 1

Answer:

To prove:

$(\sqrt{3} + \sqrt{5} )^{2}$ is an irrational

Proof:

= $(\sqrt{3} + \sqrt{5} )^{2}$

= $[(\sqrt{3})^{2} + 2 (\sqrt{3}) (\sqrt{5}) + (\sqrt{5})^{2} ]$                   Identity: $[ (a+b)^{2} = a^{2} +2ab +b^{2} ]$

= $3 + 2 \sqrt{15} + 5$

= 8  $+ 2 \sqrt{15}$

• Let us assume to the contrary that 8  $+ 2 \sqrt{15}$ is a rational number.
• Then, there exists co-prime positive integers a and b such that

        8  $+ 2 \sqrt{15}$ = $\frac{a}{b}$

    ⇒ $2 \sqrt{15}$ =  $\frac{a}{b}$ - 8

    ⇒ $2 \sqrt{15}$ = $\frac{a - 8b }{b}$

    ⇒ $\sqrt{15}$ = $\frac{a - 8b }{2b}$

• As a and b are positive integers,  $\frac{a - 8b }{2b}$ is a rational number
• Then,  $\sqrt{15}$ is also rational.
• But we know that  $\sqrt{15}$ is irrational.
• This contradicts the fact that  $\sqrt{15}$ is irrational.
• This contradiction has arisen due to our incorrect assumption that $(\sqrt{3} + \sqrt{5} )^{2}$ = 8  $+ 2 \sqrt{15}$ is rational.

Hence, we conclude that $(\sqrt{3} + \sqrt{5} )^{2}$ is an irrational number.

Hope you got that.

Thank You.

Answer 2

Step-by-step explanation:

To prove:

(\sqrt{3} + \sqrt{5} )^{2}(

3

+

5

)

2

is an irrational

Proof:

= (\sqrt{3} + \sqrt{5} )^{2}(

3

+

5

)

2

= [(\sqrt{3})^{2} + 2 (\sqrt{3}) (\sqrt{5}) + (\sqrt{5})^{2} ][(

3

)

2

+2(

3

)(

5

)+(

5

)

2

] Identity: [ (a+b)^{2} = a^{2} +2ab +b^{2} ][(a+b)

2

=a

2

+2ab+b

2

]

= 3 + 2 \sqrt{15} + 53+2

15

+5

= 8 + 2 \sqrt{15}+2

15

Let us assume to the contrary that 8 + 2 \sqrt{15}+2

15

is a rational number.

Then, there exists co-prime positive integers a and b such that

8 + 2 \sqrt{15}+2

15

= \frac{a}{b}

b

a

⇒ 2 \sqrt{15}2

15

= \frac{a}{b}

b

a

- 8

⇒ 2 \sqrt{15}2

15

= \frac{a - 8b }{b}

b

a−8b

⇒ \sqrt{15}

15

= \frac{a - 8b }{2b}

2b

a−8b

As a and b are positive integers, \frac{a - 8b }{2b}

2b

a−8b

is a rational number

Then, \sqrt{15}

15

is also rational.

But we know that \sqrt{15}

15

is irrational.

This contradicts the fact that \sqrt{15}

15

is irrational.

This contradiction has arisen due to our incorrect assumption that (\sqrt{3} + \sqrt{5} )^{2}(

3

+

5

)

2

= 8 + 2 \sqrt{15}+2

15

is rational.

Hence, we conclude that (\sqrt{3} + \sqrt{5} )^{2}(

3

+

5

)

2

is an irrational number.