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Simplify. Rationalize the denominator. \newline10−8+3\frac{10}{-8 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline10−8+3\frac{10}{-8 + \sqrt{3}}
  1. Select Conjugate: Select the conjugate of −8+3-8 + \sqrt{3}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −8+3-8 + \sqrt{3}: −8−3-8 - \sqrt{3}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the fraction by (-8 - \sqrt{3})/(-8 - \sqrt{3})\(\newline\).\newline(10/(−8+3))×((−8−3)/(−8−3))(10/(-8 + \sqrt{3})) \times ((-8 - \sqrt{3})/(-8 - \sqrt{3}))
  3. Simplify Numerator: Simplify the numerator by distributing the multiplication.\newline10×(−8−3)10 \times (-8 - \sqrt{3})\newline= 10×(−8)−10×(3)10 \times (-8) - 10 \times (\sqrt{3})\newline= −80−103-80 - 10\sqrt{3}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−8+3)∗(−8−3)(-8 + \sqrt{3}) * (-8 - \sqrt{3})\newline=(−8)2−(3)2= (-8)^2 - (\sqrt{3})^2\newline=64−3= 64 - 3\newline=61= 61
  5. Write Simplified Expression: Write the simplified expression.\newline(−80−103)/61(-80 - 10 \sqrt{3})/61\newlineThis fraction is already in simplest form.

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