If x = \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}} and y = \frac{\sqrt{7}-\sqrt{3}}{\sqrt{… — State PSC Exam

If $x = \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}-\sqrt{3}}$ and $y = \frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}+\sqrt{3}}$ , what is the value of $x^2 + y^2$ ?

Answer: 23

The value of

x^2 + y^2

is 23.

By rationalizing the denominators of

x

and

y

, we obtain

x = \frac{5+\sqrt{21}}{2}

and

y = \frac{5-\sqrt{21}}{2}

. Adding these gives

x+y=5

, and multiplying them gives

xy=1

. Using the identity

x^2+y^2 = (x+y)^2 - 2xy

, we calculate

5^2 - 2(1) = 23

Rationalize the expressions for

x

and

y

by multiplying the numerator and denominator by their respective conjugates.

x = \frac{(\sqrt{7}+\sqrt{3})^2}{7-3} = \frac{10 + 2\sqrt{21}}{4} = \frac{5 + \sqrt{21}}{2}

and

y = \frac{(\sqrt{7}-\sqrt{3})^2}{7-3} = \frac{10 - 2\sqrt{21}}{4} = \frac{5 - \sqrt{21}}{2}

Eliminating surds from the denominator simplifies addition and multiplication.

Calculate the sum

(x + y)

and the product

(x \cdot y)

x

and

y

x + y = \frac{5 + \sqrt{21}}{2} + \frac{5 - \sqrt{21}}{2} = 5

and

x \cdot y = \frac{(5 + \sqrt{21})(5 - \sqrt{21})}{4} = \frac{25 - 21}{4} = 1

Simplifying the symmetric expressions reduces computation complexity.

Apply the algebraic identity

x^2 + y^2 = (x + y)^2 - 2xy

x^2 + y^2 = (5)^2 - 2(1) = 25 - 2 = 23

Using the identity avoids evaluating complex squares directly.

Rationalization of surds and application of symmetric algebraic identities

Question