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Simplify. Rationalize the denominator. \newline10−2+3\frac{10}{-2 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline10−2+3\frac{10}{-2 + \sqrt{3}}
  1. Question Prompt: Question prompt: How can we simplify the expression and rationalize the denominator of 10−2+3\frac{10}{-2 + \sqrt{3}}?
  2. Select Conjugate: Select the conjugate of −2+3-2 + \sqrt{3}. Conjugate of a number in the form a+ba + \sqrt{b} is a−ba - \sqrt{b}, and vice versa. Conjugate of −2+3-2 + \sqrt{3} is −2−3-2 - \sqrt{3}.
  3. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline(10⋅(−2−3))/((−2+3)⋅(−2−3))(10 \cdot (-2 - \sqrt{3}))/((-2 + \sqrt{3}) \cdot (-2 - \sqrt{3}))
  4. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline10×(−2)+10×(−3)10 \times (-2) + 10 \times (-\sqrt{3})\newline= −20−103-20 - 10\sqrt{3}
  5. Simplify Denominator: Simplify the denominator using the difference of squares formula (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2.\newline(−2)2−(3)2(-2)^2 - (\sqrt{3})^2\newline=4−3= 4 - 3\newline=1= 1
  6. Write Simplified Expression: Write the simplified expression.\newline(−20−103)/1(-20 - 10\sqrt{3})/1\newlineSince dividing by 11 does not change the value, the expression simplifies to:\newline−20−103-20 - 10\sqrt{3}

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