Simplify. Rationalize the denominator. $\newline$ `(-7)/(2 + \sqrt{5}\)` $\newline$ ______

Full solution

Q. Simplify. Rationalize the denominator.

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`(-7)/(2 + \sqrt{5}\)`

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Multiply Conjugate: The conjugate of $2 + \sqrt{5}$ is $2 - \sqrt{5}$ . So we multiply both the numerator and the denominator by this conjugate. $\newline$ $\frac{(-7) \cdot (2 - \sqrt{5})}{(2 + \sqrt{5}) \cdot (2 - \sqrt{5})}$
Multiply Numerators: Now, let's multiply the numerators together: $-7 \times 2 = -14$ and $-7 \times (-\sqrt{5}) = 7\sqrt{5}$ . So the numerator becomes $-14 + 7\sqrt{5}$ .
Multiply Denominators: Next, we multiply the denominators together: $(2 + \sqrt{5}) * (2 - \sqrt{5})$ is a difference of squares, which equals $2^2 - (\sqrt{5})^2$ .
Calculate Denominator: Calculating the denominator: $2^2 = 4$ and $(\sqrt{5})^2 = 5$ . So the denominator becomes $4 - 5$ .
Simplify Denominator: The denominator simplifies to $-1$ . So the entire expression is $(-14 + 7\sqrt{5}) / -1$ .
Final Simplified Expression: Finally, we divide each term in the numerator by $-1$ to get the simplified expression. $14 - 7\sqrt{5}$ .