Question

Please solve all the three questions. Doing on pen and paper is appreciated. Thank you.​

Answer 1

Answer:

Step-by-step explanation:

8) [2⁻² * 3³ / 2³ * 3⁻⁵] ⁻⁵/²

=> [3³ * 3⁵ / 2³ * 2² ]⁻⁵/²   ( ∵ x⁻ⁿ = 1/xⁿ)

=> [3⁸/2⁵]⁻⁵/²                     (∵ xᵃ * xᵇ =  xᵃ⁺ᵇ)

=> [3⁸]⁻⁵/²   / [2⁵]⁻⁵/²    

=> 3⁻²⁰/ 2⁻²⁵/²                   (∵(xᵃ)ᵇ = xᵃᵇ)

=> 2²⁵/² / 3²⁰                    ( ∵ x⁻ⁿ = 1/xⁿ)

=> √(2²⁵)/3²⁰

=> √(2²⁴ x 2¹)/3²⁰

=> 2¹²√2/3²⁰  

7. Given x = 1 - √2.

  1/x = 1/1 - √2

    //multiply numerator and denominator by 1 + √2

=> 1/x = [1/1 - √2] [1 + √2 / 1 + √2]

          = 1 + √2 / (1 - √2) (1 + √2)

           = 1 +√2 / 1 - 2                      (∵ (a + b)(a-b) = a² - b²)

           = - (1 + √2)

Now (x - 1/x)² = {(1 - √2) - [-(1 +√2)] }²

                      = {1 - √2 + 1 + √2}²

                       = 2²

                       = 4.

6.  Given x = √5 - √2 / √5 + √2

               y =  √5 + √2/√5 - √2

x + y = [√5 - √2 / √5 + √2 ] + [√5 + √2/√5 - √2]

        = (√5 - √2)² + (√5 + √2)² /(√5 + √2)(√5 - √2)

         = 5 + 2 - 2√10 + 5 + 2 + 2√10 / (√5)² - (√2)²

          = 14/3.

xy = [√5 - √2 / √5 + √2 ] * [√5 + √2/√5 - √2] = 1

Now x² + xy + y² = (x + y)² - xy

                           = (14/3)² - 1

                           = 196/9 - 1

                           = 187/9.