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Simplify. Rationalize the denominator. $\newline$ $-\frac{10}{10 + \sqrt{3}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

-\frac{10}{10 + \sqrt{3}}

Identify Conjugate of Denominator: Identify the conjugate of the denominator. $\newline$ The conjugate of a number of the form $a + \sqrt{b}$ is $a - \sqrt{b}$ . Therefore, the conjugate of $10 + \sqrt{3}$ is $10 - \sqrt{3}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. $\newline$ $(-\frac{10}{10 + \sqrt{3}}) \times \frac{10 - \sqrt{3}}{10 - \sqrt{3}}$
Distribute Multiplication in Numerator: Distribute the multiplication in the numerator. $\newline$ Multiply $-10$ by each term in the conjugate $10 - \sqrt{3}$ . $\newline$ $-10 \times 10 = -100$ $\newline$ $-10 \times (-\sqrt{3}) = 10\sqrt{3}$ $\newline$ So, the numerator becomes $-100 + 10\sqrt{3}$ .
Expand Denominator: Expand the denominator using the difference of squares formula. $\newline$ $(10 + \sqrt{3}) \times (10 - \sqrt{3}) = 10^2 - (\sqrt{3})^2$ $\newline$ $100 - 3 = 97$ $\newline$ So, the denominator becomes $97$ .
Write Simplified Expression: Write the simplified expression. $\newline$ The simplified expression with the rationalized denominator is $(-100 + 10\sqrt{3})/97$ .

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