Question

Which expressions is below equal a rational number? Choose all that apply

Answer 1

A,D,E

Explanation:

Rational number :That number which can be written as
`(p)/(q)` where q is not equal to zero and p,q are both integers.

We have to find which number is rational number

1.
`6\sqrt 3-√(108)`

`6\sqrt3-√(6* 6* 3)`

`6\sqrt3-6\sqrt3=0`

0 is rational number by define of rational number .

Hence, option A is true.

B.
`\pi\sqrt9`

We know that
`\pi` is a irrational number when we multiply a irrational number with rational number then the number becomes irrational number

`\pi√(3* 3)`

`3\pi`

Where 3 is rational number and
`\pi`is a irrational number

Hence, the number
`3\pi` is a irrational number not rational number.

Therefore, option B is not true.

C.
`√(49)+\sqrt5`

`√(7* 7)-\sqrt5`

`7-\sqrt5`

We know that
`\sqrt5` is a irrational number

When we add rational number to irrational number then the number becomes irrational number .

So,
`7-\sqrt5` is a irrational number.

Hence, option C is not true.

D.
`(5\sqrt3)(4\sqrt3)`

`20(\sqrt3)^2`

`20* 3`

=60

60 is a rational number because can be written as
`(p)/(q)` form.

Hence, option D is true.

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

`(5+12)/(15)`

`(17)/(15)`

It is a rational number because it is in the form of
`(p)/(q)`.

Hence, option E is true.

Answer 2

`6√(3) -√(108)`

`(5√(3))(4√(3))`

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

Explanation:

we know that

A rational number is a number that can be written as a ratio of two integers

so

Verify each case

case A)
`6√(3) -√(108)`

we know that

`√(108)=6√(3)`

substitute

`6√(3) -√(108)=6√(3)-6√(3)=0`

The number 0 is a rational number, because can be written as a ratio of two integers

Example
`(0)/(1)=0`

therefore

The expression is a rational number

case B)
`\pi √(9)`

`\pi √(9)=3\pi`

Is not a rational number, because cannot be written as a ratio of two integers

case C)
`√(49)+√(5)`

`√(49)+√(5)=7+√(5)`

Is not a rational number, because cannot be written as a ratio of two integers

case D)
`(5√(3))(4√(3))`

`(5√(3))(4√(3))=20*3=60`

The number 60 is a rational number, because can be written as a ratio of two integers

Example
`(120)/(2)=60`

therefore

The expression is a rational number

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

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

The number is a rational number, because can be written as a ratio of two integers

therefore

The expression is a rational number