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Use the imaginary number ii to rewrite the expression below as a complex number. Simplify all radicals. −72\sqrt{-72}

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Full solution

Q. Use the imaginary number ii to rewrite the expression below as a complex number. Simplify all radicals. −72\sqrt{-72}
  1. Express as Product: Express −72\sqrt{-72} as the product of square roots and −1\sqrt{-1}.\newline−72=−1×72\sqrt{-72} = \sqrt{-1 \times 72}
  2. Break Down Prime Factors: Break down 7272 into its prime factors to simplify the radical.\newline72=2×2×2×3×3=23×3272 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2
  3. Separate Perfect Squares: Separate the perfect squares from the prime factorization. 23×32=22×32×2\sqrt{2^3 \times 3^2} = \sqrt{2^2 \times 3^2 \times 2}
  4. Simplify Square Roots: Simplify the square roots of the perfect squares and pull them out of the radical. 22×32×2=2×3×2=6×2\sqrt{2^2 \times 3^2 \times 2} = 2 \times 3 \times \sqrt{2} = 6 \times \sqrt{2}
  5. Express as Imaginary Unit: Express −1\sqrt{-1} as the imaginary unit ii.−1=i\sqrt{-1} = i
  6. Combine to Form Complex Number: Combine the imaginary unit with the simplified radical to form the complex number. i×6×2=6i×2i \times 6 \times \sqrt{2} = 6i \times \sqrt{2}

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