Question

jika
[tex] {x}^{ \frac{1}{2} } + {x}^{ - \frac { 1}{2} } = 3[/tex]
maka nilai dari
[tex]x + {x}^{ - 1} [/tex]
adalah​

Answer 1

[tex]\begin{aligned}&\textsf{Jika }x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3,\\&\textsf{maka nilai dari }x+x^{-1}\\&\sf adalah\ \boxed{\,\bf7\,}\ .\end{aligned}[/tex]
 

Penjelasan dengan langkah-langkah:

Eksponen

Diketahui:

[tex]\begin{aligned}x^{\frac{1}{2}} + x^{-\frac{1}{2}} = 3\end{aligned}[/tex]

Ditanyakan:

[tex]x+x^{-1}=...?[/tex]

Penyelesaian

Ingat a² + b² = (a + b)² – 2ab.

Maka:

[tex]\begin{aligned}x+x^{-1}&=\left(x^{\frac{1}{2}}\right)^2 + \left(x^{-\frac{1}{2}}\right)^2\\&=\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2-2x^{\frac{1}{2}}x^{-\frac{1}{2}}\\&=\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2-2x^{\frac{1}{2}-\frac{1}{2}}\\&=\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2-2x^0\\&=\left(x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right)^2-2\\&=3^2-2\\x+x^{-1}&=\boxed{\,\bf7\,}\quad\blacksquare\end{aligned}[/tex]