Question

Examine whether the following numbers are rational or irrational with justification

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

$(2 + \sqrt{2})(2 - \sqrt{2})$
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Answer 1

Step-by-step explanation:

Given :-

(3+√5)^2 and (2+√2)(2-√2)

To find:-

Examine whether the following numbers are rational or irrational with justification.

Solution:-

1)Given number = (3+√5)^2

It is in the form of (a+b)^2

Where a = 3 and b=√5

(a+b)^2 = a^2+2ab+b^2

(3+√5)^2

=> 3^2+2(3)(√5)+(√5)^2

=>9+6√5+5

=> 14+6√5

(3+√5)^2 = 14+6√5

(3+√5)^2 is an irrational number.

2) Given number = (2+√2)(2-√2)

It is in the form of (a+b)(a-b)

Where a = 2 and b=√2

We know that

(a+b)(a-b)=a^2-b^2

(2+√2)(2-√2) = 2^2 - (√2)^2

=> 4-2

=> 2

(2+√2)(2-√2) = 2

(2+√2)(2-√2) is a rational number.

Answer:-

1)(3+√2)^2 is an irrational number

2)(2+√2)(2-√2) is a rational number.

Used formula:-

• (a+b)^2 = a^2+2ab+b^2

• (a+b)(a-b)=a^2-b^2

• q is a rational number and s is an irrational number then q+s , q-s,qs and q/s are also irrational numbers.

Answer 2

Step-by-step explanation:

hope this helps you

and you get your answer