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Simplify. Rationalize the denominator.\newline8−5−2\frac{8}{-5 - \sqrt{2}}

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Full solution

Q. Simplify. Rationalize the denominator.\newline8−5−2\frac{8}{-5 - \sqrt{2}}
  1. Find Conjugate: Select the conjugate of −5−2-5 - \sqrt{2}.\newlineConjugate of a−ba - \sqrt{b}: a+ba + \sqrt{b}\newlineConjugate of −5−2-5 - \sqrt{2}: −5+2-5 + \sqrt{2}
  2. Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator.\newlineTo rationalize the denominator, we multiply the expression by the conjugate of the denominator over itself, which is 11.\newline8⋅(−5+2)(−5−2)⋅(−5+2)\frac{8 \cdot (-5 + \sqrt{2})}{(-5 - \sqrt{2}) \cdot (-5 + \sqrt{2})}
  3. Simplify Numerator: Simplify the numerator.\newline8×(−5+2)=8×−5+8×2=−40+8×28 \times (-5 + \sqrt{2}) = 8 \times -5 + 8 \times \sqrt{2} = -40 + 8 \times \sqrt{2}
  4. Simplify Denominator: Simplify the denominator using the difference of squares formula.\newline(−5−2)∗(−5+2)=(−5)2−(2)2=25−2=23(-5 - \sqrt{2}) * (-5 + \sqrt{2}) = (-5)^2 - (\sqrt{2})^2 = 25 - 2 = 23
  5. Write Simplified Expression: Write the simplified expression.\newlineThe simplified expression is (−40+8×2)/23(-40 + 8 \times \sqrt{2}) / 23

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