Question

Which expressions below equal a rational number

Answer 1

6√3 - √108
(5√3) (4√3)

`(1)/(3)` +
`(4)/(5)`

Those are all rational numbers.

Answer 2

we know that

A rational number is any number that can be expressed as a ratio of two integers.

case A)
`6√(3)- √(108)`

we know that

`108=2^(2)3^(3)`

so

`√(108)=\sqrt{2^(2)3^(3)}=6√(3)`

substitute in the expression

`6√(3)- 6√(3)=0`

The number
`0` is a rational number since it can be written as
`0/2`

case B)
`\pi √(9)`

we know that

`\pi √(9)=3 \pi`

The number
`3 \pi` is not a rational number because it can't be expressed as a relationship of two integers

case C)
`√(49)+ √(5)`

we know that

`√(49)+ √(5)=7+√(5)`

The number
`7+√(5)` is not a rational number because it can't be expressed as a relationship of two integers

case D)
`(5√(3))(4√(3))`

we know that

`(5√(3))(4√(3))=(5)(4)(√(3))(√(3))=(20)(3)=60`

The number
`60` is a rational number since it can be written as
`60/1`

case E)
`(1)/(3) +(4)/(5)`

we know that

`(1)/(3) +(4)/(5)=(5*1+3*4)/(5*3)= (17)/(15)`

The number
`(17)/(15)` is a rational number since it can be expressed as a ratio of two integers

therefore

the answer is

`6√(3)- √(108)`

`(5√(3))(4√(3))`

`(5√(3))(4√(3))`