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Simplify. Rationalize the denominator. \newline104+3\frac{10}{4 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator. \newline104+3\frac{10}{4 + \sqrt{3}}
  1. Select Conjugate: Select the conjugate of 4+34 + \sqrt{3}.\newlineConjugate of a+ba + \sqrt{b}: a−ba - \sqrt{b}\newlineConjugate of 4+34 + \sqrt{3}: 4−34 - \sqrt{3}
  2. Multiply by Conjugate: Multiply the original expression by the conjugate over itself to rationalize the denominator.\newline(104+3)×4−34−3(\frac{10}{4 + \sqrt{3}}) \times \frac{4 - \sqrt{3}}{4 - \sqrt{3}}
  3. Simplify Numerator: Simplify the numerator by distributing the 1010 to both terms in the conjugate.10×(4−3)10 \times (4 - \sqrt{3})=10×4−10×3= 10 \times 4 - 10 \times \sqrt{3}=40−10×3= 40 - 10 \times \sqrt{3}
  4. Simplify Denominator: Simplify the denominator by using the difference of squares formula.\newline(4+3)×(4−3)(4 + \sqrt{3}) \times (4 - \sqrt{3})\newline=42−(3)2= 4^2 - (\sqrt{3})^2\newline=16−3= 16 - 3\newline=13= 13
  5. Write Simplified Expression: Write the simplified expression with the rationalized denominator.\newline(40−10⋅3)/13(40 - 10 \cdot \sqrt{3}) / 13\newlineThis fraction is already in simplest form.

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