Question

What is the simplest form of the expression sqrt 2-sqrt 10/sqrt 2+sqrt 10

Answer 1

To simplify the expression sqrt 2-sqrt 10/sqrt 2+sqrt 10, we need to rationalize the denominator by multiplying both the numerator and denominator by sqrt(2)-sqrt(10). This results in the expression (-3+sqrt(5))/(-2).

Step-by-step explanation:

To find the simplest form of the expression, we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

The conjugate of sqrt(2)+sqrt(10) is sqrt(2)-sqrt(10).

Multiplying the numerator and denominator by sqrt(2)-sqrt(10) gives us:

((sqrt(2)-sqrt(10))*(sqrt(2)-sqrt(10))) / ((sqrt(2)+sqrt(10))*(sqrt(2)-sqrt(10)))

This simplifies to:

((sqrt(2))^2-2*sqrt(2)*sqrt(10)+(sqrt(10))^2)/(2-10)

Simplifying the numerator and denominator gives us:

(2-2*sqrt(20)+10)/(-8)

(12-2*sqrt(20))/(-8)

Simplifying further:

(-6+sqrt(20))/(-4)

This can be further simplified by dividing the numerator and denominator by 2:

(-3+sqrt(5))/(-2)

The simplest form of the expression is (-3+sqrt(5))/(-2).

Answer 2

The simplest form of given expression
`(\sqrt 2-\sqrt 10)/( \sqrt 2+\sqrt 10)` is
`-(3+√(5))/(2)`

Step-by-step explanation:

Given expression
`(\sqrt 2-\sqrt 10)/( \sqrt 2+\sqrt 10)`

We are required to simplify the above given expression,

Consider the given expression,
`(\sqrt 2-\sqrt 10)/( \sqrt 2+\sqrt 10)`

We will first rationalize the denominator by multiply and divide by
`{\sqrt 2-\sqrt 10}`

Then , given expression becomes,

`\Rightarrow (\sqrt 2-\sqrt 10)/( \sqrt 2+\sqrt 10)* (\sqrt 2-\sqrt 10)/(\sqrt 2-\sqrt 10)`

Using identity
`(a+b)(a-b)=a^2-b^2` in denominator, we get,

`\Rightarrow ((\sqrt 2-\sqrt 10)( \sqrt 2-\sqrt 10))/(2-10)`

`\Rightarrow ((\sqrt 2-\sqrt 10)( \sqrt 2-\sqrt 10))/(-8)`

In numerator, Using identity
`(a-b)(a-b)=(a-b)^2`, we get,

`\Rightarrow ((\sqrt 2-\sqrt 10)^2)/(-8)`

Using
`(a-b)^2=a^2+b^2-2ab` , we get,

`\Rightarrow ( 2+10-4√(20))/(-8)`

`\Rightarrow (12+4√(5))/(-8)`

On simplifying, we get,

`(12+4√(5))/(-8)=-(3+√(5))/(2)`

Thus, the simplest form of given expression
`(\sqrt 2-\sqrt 10)/( \sqrt 2+\sqrt 10)` is
`-(3+√(5))/(2)`