Question

Which of the following are irrational?

विकल्प

• `sqrt(80)/sqrt(5)`

• `(sqrt(7) - sqrt(3))^2`

• `(sqrt(50) + sqrt(8))^2`

• `sqrt(288)/sqrt(2)`

• `(3sqrt(80) - 6sqrt(45))^2`

Answer 1

`bb((sqrt(7) - sqrt(3))^2)`

Explanation:

a. `sqrt(80)/sqrt(5)`

We can simplify this:

`sqrt(80)/sqrt(5)`

= `sqrt(80/5)`

= `sqrt(16)`

= 4

Since 4 is a rational number, the result is rational.

b. `(sqrt(7) - sqrt(3))^2`

Let’s expand this expression using the identity (a – b)² = a² – 2ab + b²:

`(sqrt(7) - sqrt(3))^2`

= `(sqrt(7))^2 - 2(sqrt(7))(sqrt(3)) + (sqrt(3))^2`

= `7 - 2sqrt(21) + 3`

= `10 - 2sqrt(21)`

Since `sqrt(21)` is irrational, `10 - 2sqrt(21)` is irrational.

c. `(sqrt(50) + sqrt(8))^2`

Let’s expand this expression:

`(sqrt(50) + sqrt(8))^2`

= `(sqrt(50))^2 + 2(sqrt(50))(sqrt(8)) + (sqrt(8))^2`

= `50 + 2sqrt(400) + 8`

= 50 + 40 + 8

= 98

Since 98 is a rational number, the result is rational.

d. `sqrt(288)/sqrt(2)`

We can simplify this:

`sqrt(288)/sqrt(2)`

= `sqrt(288/2)`

= `sqrt(144)`

= 12

Since 12 is a rational number, the result is rational.

e. `(3sqrt(80) - 6sqrt(45))^2`

We can simplify this:

`(3sqrt(80) - 6sqrt(45))^2`

= `(3(4sqrt(5)) - 6(3sqrt(5))^2`

= `(12sqrt(5) - 18sqrt(5))^2`

= `(-6sqrt(5))^2`

= `(-6)^2 xx (sqrt(5))^2`

= 36 × 5

= 180

Since 180 is a rational number, the result is rational.