Question

Which of the following is equivalent to the radical expression below, when the denominator has been rationalized and X >_ 5?

Answer 1

The correct option is D)
`2\left(√(x)+√(x-5)\right)`.

Explanation:

Consider the provided radical expression.

`(10)/(√(x)-√(x-5))`

Multiply by the conjugate
`(√(x)+√(x-5))/(√(x)+√(x-5))`

`(10\left(√(x)+√(x-5)\right))/(\left(√(x)-√(x-5)\right)\left(√(x)+√(x-5)\right))`

`(10\left(√(x)+√(x-5)\right))/(\left(x-(x-5)))`

`(10\left(√(x)+√(x-5)\right))/(5)`

`2\left(√(x)+√(x-5)\right)`

Hence, the correct option is D)
`2\left(√(x)+√(x-5)\right)`.

Answer 2

Option D is correct.

Explanation:

`(10)/(√(x)-√(x-5))\\` We need to rationalize this term and find the answer.

To rationalize the term we multiply and divide the above expression by
`√(x)+√(x-5)`

Solving:

`(10)/(√(x)-√(x-5))\\=(10)/(√(x)-√(x-5)) * (√(x)+√(x-5))/(√(x)+√(x-5)) \\Multiplying\\=(10*(√(x)+√(x-5)))/(√(x)-√(x-5)*√(x)+√(x-5))\\=(10*(√(x)+√(x-5)))/((√(x))^2-(√(x-5))^2)\\=(10*(√(x)+√(x-5)))/(x-(x-5))\\=(10*(√(x)+√(x-5)))/(x-x+5))\\=(10*(√(x)+√(x-5)))/(5)\\=2*(√(x)+√(x-5))`

So, Option D is correct.