Pregunta
Simplify: \( \frac{3-\sqrt{5}}{2+\sqrt{5}} \)
Ask by Simmons Matthews. in the United States
Jan 24,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
\( 5\sqrt{5} - 11 \)
Solución
To simplify the expression:
\[
\frac{3 - \sqrt{5}}{2 + \sqrt{5}}
\]
we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is \(2 - \sqrt{5}\):
\[
\frac{3 - \sqrt{5}}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{(3 - \sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}
\]
**Step 1: Expand the Numerator**
\[
(3 - \sqrt{5})(2 - \sqrt{5}) = 3 \cdot 2 + 3 \cdot (-\sqrt{5}) + (-\sqrt{5}) \cdot 2 + (-\sqrt{5}) \cdot (-\sqrt{5})
\]
\[
= 6 - 3\sqrt{5} - 2\sqrt{5} + 5
\]
\[
= 11 - 5\sqrt{5}
\]
**Step 2: Simplify the Denominator**
\[
(2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
\]
**Step 3: Combine the Results**
\[
\frac{11 - 5\sqrt{5}}{-1} = -11 + 5\sqrt{5} = 5\sqrt{5} - 11
\]
**Final Answer:**
\[
5\sqrt{5} - 11
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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To simplify \( \frac{3-\sqrt{5}}{2+\sqrt{5}} \), we start by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( 2 - \sqrt{5} \): \[ \frac{(3 - \sqrt{5})(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})} \] Calculating the denominator: \[ (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \] Now, calculating the numerator: \[ (3 - \sqrt{5})(2 - \sqrt{5}) = 3 \cdot 2 - 3 \cdot \sqrt{5} - \sqrt{5} \cdot 2 + (\sqrt{5})^2 \] \[ = 6 - 3\sqrt{5} - 2\sqrt{5} + 5 = 11 - 5\sqrt{5} \] Putting it all together, we get: \[ \frac{11 - 5\sqrt{5}}{-1} = -11 + 5\sqrt{5} \] Thus, the simplified expression is: \[ 5\sqrt{5} - 11 \]
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