If x = \sqrt{3} + \sqrt{2}, what is the value of x^4 + x^{-4}? — GMAT

If $x = \sqrt{3} + \sqrt{2}$ , what is the value of $x^4 + x^{-4}$ ?

Rationalizing the denominator gives

x^{-1} = \sqrt{3} - \sqrt{2}

. Summing

x + x^{-1}

yields

2\sqrt{3}

. Squaring this sum gives

x^2 + 2 + x^{-2} = 12

, so

x^2 + x^{-2} = 10

. Squaring once more gives

x^4 + 2 + x^{-4} = 100

, which simplifies to

x^4 + x^{-4} = 98

Find the reciprocal expression

x^{-1}

x^{-1} = \frac{1}{\sqrt{3} + \sqrt{2}} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \sqrt{3} - \sqrt{2}

Rationalizing the denominator simplifies the expression for

x^{-1}

Calculate the sum

x + x^{-1}

x + x^{-1} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3}

Adding the original expression and its reciprocal eliminates the

\sqrt{2}

term.

Square

x + x^{-1}

to find

x^2 + x^{-2}

(x + x^{-1})^2 = x^2 + 2(x)(x^{-1}) + x^{-2} = x^2 + 2 + x^{-2} = (2\sqrt{3})^2 = 12 \implies x^2 + x^{-2} = 10

Using the algebraic identity

(a+b)^2 = a^2 + 2ab + b^2

allows finding

x^2 + x^{-2}

Square

x^2 + x^{-2}

to find

x^4 + x^{-4}

(x^2 + x^{-2})^2 = x^4 + 2(x^2)(x^{-2}) + x^{-4} = x^4 + 2 + x^{-4} = 10^2 = 100 \implies x^4 + x^{-4} = 98

Squaring the second-power sum and subtracting the cross-term 2 yields the required value.

Rationalizing radical denominators and binomial squaring of reciprocals

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