Question

Ayuda ):
Calcular el valor de (x+y)²+ (x-y)², si se sabe que: x+1 = √2; y-1=√2
Si : (a+1/a)³=27, hallar: a³ + 1/a³
Si: x/y + y/x = 7 , calcular: √x/y + √y/x

Answer 1

Explicación paso a paso:

Calcular el valor de (x+y)²+ (x-y)²

Calcular el valor de (x+y)²+ (x-y)²si se sabe que: x+1 = √2; y-1=√2

$x + 1 = \sqrt{2}$

$x = \sqrt{2} - 1$

$y - 1 = \sqrt{2}$

$y = \sqrt{2} + 1$

Reemplaza en

$(x+y)²+ (x-y)²$

${( \sqrt{2 } - 1 + \sqrt{2} + 1 })^{2} + {( \sqrt{2} - 1 - ( \sqrt{2} + 1) })^{2}$

${(2 \sqrt{2} })^{2} - {( - 2)}^{2}$

$8 - 4 = 4$

Si : (a+1/a)³=27, hallar: a³ + 1/a³

$(a+1/a)³=27$

$a+1/a=3$

$Si : (a+1/a)³=27$

${a}^{3} + 3 {a}^{2} ( \frac{1}{a} ) + 3a( { \frac{1}{a} })^{2} + { (\frac{1}{a} })^{3} = 27$

${a}^{3} + 3a + \frac{3}{a} + {a}^{ - 3} = 27$

Entonces: a³ + 1/a³

$a³ + 1/a³ = 27 - 3(a + {a}^{ - 1} )$

${a}^{3} + {a }^{ - 3} = 27 - 3(3)$

${a}^{3} + {a }^{ - 3} = 18$

Si: x/y + y/x = 7 , calcular: √x/y + √y/x

$\frac{x}{y} + \frac{y}{x} = 7$

$\sqrt{ \frac{x}{y} } + \sqrt{ \frac{y}{x} }$

$\frac{ {x}^{2} + {y}^{2} }{xy} = 7$

${x}^{2} + {y}^{2} = 7xy$

${x}^{2} + 2xy + {y}^{2} = 9xy$

${(x + y)}^{2} = 9xy$

$x + y = \sqrt{9xy}$

$x + y = 3 \sqrt{xy}$

Entonces reemplaza:

$\frac{ \sqrt{x} }{ \sqrt{y} } + \frac{ \sqrt{y} }{ \sqrt{x} } = \frac{x + y}{ \sqrt{xy} }$

$\frac{x + y}{ \sqrt{xy} } = \frac{3 \sqrt{xy} }{ \sqrt{xy} }$

Simplificación

$resp = 3$