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Simplify. Rationalize the denominator.\newline88+3\frac{8}{8 + \sqrt{3}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline88+3\frac{8}{8 + \sqrt{3}}
  1. Identify Conjugate of Denominator: Identify the conjugate of the denominator.\newlineThe conjugate of a number of the form a+ba + \sqrt{b} is a−ba - \sqrt{b}. Therefore, the conjugate of 8+38 + \sqrt{3} is 8−38 - \sqrt{3}.
  2. Multiply by Conjugate Fraction: Multiply the original expression by a fraction equivalent to 11 that has the conjugate of the denominator as both its numerator and denominator.\newline(88+3)×8−38−3=8×(8−3)(8+3)×(8−3)(\frac{8}{8 + \sqrt{3}}) \times \frac{8 - \sqrt{3}}{8 - \sqrt{3}} = \frac{8 \times (8 - \sqrt{3})}{(8 + \sqrt{3}) \times (8 - \sqrt{3})}
  3. Multiply Numerators: Multiply the numerators together.\newline8×(8−3)=64−838 \times (8 - \sqrt{3}) = 64 - 8\sqrt{3}
  4. Multiply Denominators: Multiply the denominators together using the difference of squares formula, a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b). \newline(8+3)×(8−3)=82−(3)2=64−3(8 + \sqrt{3}) \times (8 - \sqrt{3}) = 8^2 - (\sqrt{3})^2 = 64 - 3
  5. Simplify Denominator: Simplify the denominator. 64−3=6164 - 3 = 61
  6. Write Simplified Expression: Write the simplified expression. (64−83)/61(64 - 8\sqrt{3})/61

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