If x > 0 and x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3, what is the value of x^2 + x^{-2}? — GRE General Test

If $x > 0$ and $x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3$ , what is the value of $x^2 + x^{-2}$ ?

Squaring both sides of

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

gives

x + 2 + x^{-1} = 9

, which simplifies to

x + x^{-1} = 7

. Squaring both sides of

x + x^{-1} = 7

gives

x^2 + 2 + x^{-2} = 49

, which yields

x^2 + x^{-2} = 47

Square both sides of the given equation

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

(x^{\frac{1}{2}} + x^{-\frac{1}{2}})^2 = 3^2 \implies x + 2(x^{\frac{1}{2}})(x^{-\frac{1}{2}}) + x^{-1} = 9

Applying the binomial expansion identity

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

Simplify the middle term and solve for

x + x^{-1}

x + 2(1) + x^{-1} = 9 \implies x + x^{-1} = 7

Since

x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}} = x^0 = 1

, subtracting 2 from both sides isolates

x + x^{-1}

Square both sides of

x + x^{-1} = 7

(x + x^{-1})^2 = 7^2 \implies x^2 + 2(x)(x^{-1}) + x^{-2} = 49

Squaring

x + x^{-1}

generates the terms

x^2

and

x^{-2}

Simplify the middle term and solve for

x^2 + x^{-2}

x^2 + 2 + x^{-2} = 49 \implies x^2 + x^{-2} = 47

Subtracting 2 from both sides isolates the desired expression

x^2 + x^{-2}

Algebraic Exponents and Binomial Expansion

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