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Simplify. Rationalize the denominator. \newline8−2−3\frac{8}{-2 - \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline8−2−3\frac{8}{-2 - \sqrt{3}}
  1. Select Conjugate: Select the conjugate of −2−3-2 - \sqrt{3}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −2−3-2 - \sqrt{3}: −2+3-2 + \sqrt{3}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator to rationalize it.\newline(8)/(−2−3)⋅(−2+3)/(−2+3)(8) / (-2 - \sqrt{3}) \cdot (-2 + \sqrt{3}) / (-2 + \sqrt{3})
  3. Apply Distributive Property: Apply the distributive property to multiply the numerators and the denominators.\newlineNumerator: 8×(−2+3)=8×−2+8×3=−16+838 \times (-2 + \sqrt{3}) = 8 \times -2 + 8 \times \sqrt{3} = -16 + 8\sqrt{3}\newlineDenominator: (−2−3)×(−2+3)=(−2)2−(3)2=4−3(-2 - \sqrt{3}) \times (-2 + \sqrt{3}) = (-2)^2 - (\sqrt{3})^2 = 4 - 3
  4. Simplify Denominator: Simplify the denominator. 4−3=14 - 3 = 1
  5. Final Answer: Since the denominator is now 11, the simplified form of the expression is just the numerator.\newlineFinal Answer: −16+83-16 + 8\sqrt{3}

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