1977 AMC 12 Problema 7

Question

Si

t=\dfrac1{1-\sqrt[4]{2}},

entonces

t

es igual a

If

t=\dfrac1{1-\sqrt[4]{2}},

then

t

equals

(1-\sqrt[4]{2})(2-\sqrt2)

(1-\sqrt[4]{2})(1+\sqrt2)

(1+\sqrt[4]{2})(1-\sqrt2)

(1+\sqrt[4]{2})(1+\sqrt2)

-(1+\sqrt[4]{2})(1+\sqrt2)

Sea

u=\sqrt[4]{2}

y usa

u^4=2

Let

u=\sqrt[4]{2}

and use

u^4=2

Factoriza

u^4-1

como producto de dos factores lineales y uno cuadrático

Factor

u^4-1

into a linear factor, a second linear factor, and a quadratic factor

Sea

u=\sqrt[4]{2}.

Como

\begin{aligned} &(1-u)\bigl(-(1+u)(1+u^2)\bigr)\\ &\qquad=-(1-u^2)(1+u^2)\\ &\qquad=u^4-1=1, \end{aligned}

se sigue que

t=-(1+\sqrt[4]{2})(1+\sqrt2).

Por lo tanto, la respuesta correcta es E.

Let

u=\sqrt[4]{2}.

Since

\begin{aligned} &(1-u)\bigl(-(1+u)(1+u^2)\bigr)\\ &\qquad=-(1-u^2)(1+u^2)\\ &\qquad=u^4-1=1, \end{aligned}

it follows that

t=-(1+\sqrt[4]{2})(1+\sqrt2).

Therefore, the correct answer is E.

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