Question

The square of the sum of a real number and its reciprocal is 5. a) Determine the sum of the square of the number and the reciprocal of its square without calculating the number. b) Determine the sum of the cube and the reciprocal of the number without calculating the number.

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Answer to a math question The square of the sum of a real number and its reciprocal is 5. a) Determine the sum of the square of the number and the reciprocal of its square without calculating the number. b) Determine the sum of the cube and the reciprocal of the number without calculating the number.

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Seamus
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98 Answers
a) Legyen x a valós szám. A reciprokának az adott számnak szoktuk az x^{-1} jelölést használni. A feltétel alapján tudjuk, hogy
x + \frac{1}{x} = \sqrt{5}.
Az a) részfeladatban azt kell meghatároznunk, hogy x^2 + \left( \frac{1}{x} \right)^2 . Ennek a kifejezésnek a kiszámításához először nézzük meg, hogy mi történik, ha négyzetre emeljük a fenti egyenletet:
(x + \frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2} = 5.
Innen már könnyen meghatározható az x^2 + \left( \frac{1}{x} \right)^2 értéke:
x^2 + \left( \frac{1}{x} \right)^2 = 5 - 2 = 3.

b) Az adott feltétel alapján tudjuk, hogy
x^3 + \frac{1}{x^3} = \left( x + \frac{1}{x} \right)^3 - 3 \left( x + \frac{1}{x} \right) = \sqrt{5}^3 - 3 \cdot \sqrt{5} = 5 \sqrt{5} - 3 \sqrt{5} = 2 \sqrt{5}.
Tehát a kért szám az x^3 + \left( \frac{1}{x} \right)^3 kifejezés értéke 2 \sqrt{5} .

**Válasz:**

a) x^2 + \left( \frac{1}{x} \right)^2 = 3

b) x^3 + \left( \frac{1}{x} \right)^3 = 2 \sqrt{5}

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