# If X = √ 5 + √ 3 √ 5 − √ 3 and Y = √ 5 − √ 3 √ 5 + √ 3 Then X + Y +Xy= - Mathematics

Advertisement Remove all ads
Advertisement Remove all ads
Advertisement Remove all ads
MCQ

If $x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ and $y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ then x + y +xy=

#### Options

• 9

• 5

• 17

• 7

Advertisement Remove all ads

#### Solution

Given that  x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3)and y = (sqrt5 - sqrt3)/(sqrt5 +sqrt3).

We are asked to find  x+y + xy

Now we will rationalize x. We know that rationalization factor for  sqrt5 -sqrt3 is sqrt5 +sqrt3. We will multiply numerator and denominator of the given expression   x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3) by sqrt5 + sqrt3, to get

x=(sqrt5 +sqrt3)/(sqrt5 - sqrt3) xx (sqrt5 +sqrt3)/(sqrt5 + sqrt3)

= ((sqrt5)^2+(sqrt3)^2+ 2 xx sqrt5 xx sqrt3)/((sqrt5)^2 - (sqrt3)^2)

= (5+3+2sqrt15)/(5-3)

= 4 + sqrt15

Similarly, we can rationalize y. We know that rationalization factor for   sqrt5 +sqrt3issqrt5 - sqrt3. We will multiply numerator and denominator of the given expression   (sqrt5 - sqrt3)/(sqrt5+sqrt3)by,sqrt5 - sqrt3 to get

x = (sqrt5 - sqrt3)/(sqrt5+sqrt3) xx (sqrt5 - sqrt3)/(sqrt5-sqrt3)

 = ((sqrt5)^2 + (sqrt3)^2 - 2 xx sqrt5 xx sqrt3)/((sqrt5)^2 - (sqrt3))

= (5+3-2sqrt15)/(5-3)

= (8-2sqrt15)/2

= 4-sqrt15

Therefore,

x+y+xy = 4 +sqrt15 + 4 -sqrt15 +(4+sqrt15) (4-sqrt15)

= 4+4 + 16 - 4sqrt15 + 4 sqrt15 - (sqrt15)^2

 = 24 - 15

= 9

Concept: Laws of Exponents for Real Numbers
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 3 Rationalisation
Exercise 3.4 | Q 13 | Page 17

#### Video TutorialsVIEW ALL [1]

Share
Notifications

View all notifications

Forgot password?