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Use the imaginary number ii to rewrite the expression below as a complex number. Simplify all radicals. −68\sqrt{-68}

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Full solution

Q. Use the imaginary number ii to rewrite the expression below as a complex number. Simplify all radicals. −68\sqrt{-68}
  1. Split into Factors: Express −68\sqrt{-68} as the product of square roots and −1\sqrt{-1}.\newline−68=−1×68\sqrt{-68} = \sqrt{-1 \times 68}
  2. Express as Complex Number: Express −1×68\sqrt{-1 \times 68} as a complex number using ii.\newline−68=−1×68\sqrt{-68} = \sqrt{-1} \times \sqrt{68}
  3. Find Prime Factors: Simplify 68\sqrt{68} by finding the prime factors of 6868.68=2×2×1768 = 2 \times 2 \times 17
  4. Express as Square Root: Express 68\sqrt{68} as 2×2×17\sqrt{2 \times 2 \times 17}.\newline68=4×17\sqrt{68} = \sqrt{4 \times 17}
  5. Simplify Square Root: Simplify 4×17\sqrt{4 \times 17} to 2×172 \times \sqrt{17}.\newline68=2×17\sqrt{68} = 2 \times \sqrt{17}
  6. Combine Results: Combine the results to express the original expression as a complex number. −68=−1×2×17=2i×17\sqrt{-68} = \sqrt{-1} \times 2 \times \sqrt{17} = 2i \times \sqrt{17}
  7. Final Simplification: Simplify the expression to its final form. −68=2i⋅17=2i⋅17\sqrt{-68} = 2i \cdot \sqrt{17} = 2i \cdot \sqrt{17}

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