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Express the radical using the imaginary unit, i. Express your answer in simplified form. +-sqrt(-35)=+-

Express the radical using the imaginary unit, i i .\newlineExpress your answer in simplified form.\newline±−35=± \pm \sqrt{-35}= \pm

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Full solution

Q. Express the radical using the imaginary unit, i i .\newlineExpress your answer in simplified form.\newline±−35=± \pm \sqrt{-35}= \pm
  1. Recognizing the imaginary unit: First, we recognize that the square root of a negative number involves the imaginary unit ii, where i2=−1i^2 = -1. We can rewrite the expression ±−35\pm\sqrt{-35} by factoring out −1-1 from under the radical to separate the real and imaginary parts.\newline±−35=±−1⋅35\pm\sqrt{-35} = \pm\sqrt{-1 \cdot 35}
  2. Replacing −1\sqrt{-1} with ii: Next, we know that −1\sqrt{-1} is the definition of the imaginary unit ii. So we can replace −1\sqrt{-1} with ii and continue simplifying the expression.\newline±−1×35=±i×35\pm\sqrt{-1 \times 35} = \pm i \times \sqrt{35}
  3. Simplifying the expression: Since 3535 is a product of 55 and 77, and neither 55 nor 77 has a square root that is an integer, we cannot simplify the radical any further. Therefore, the expression is already in its simplest form.\newline±i⋅35=±i35\pm i \cdot \sqrt{35} = \pm i\sqrt{35}

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