Math Problem Statement

Express
√ − 5 as a complex number, in terms of i: 5

√ − 5

Solution

To express 5\sqrt{-5} as a complex number, we can rewrite it in terms of ii, the imaginary unit, where i=1i = \sqrt{-1}.

  1. Start with the expression:
    5\sqrt{-5}

  2. Rewrite 5-5 as 15-1 \cdot 5: 5=15\sqrt{-5} = \sqrt{-1 \cdot 5}

  3. Use the property ab=ab\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}: 5=15\sqrt{-5} = \sqrt{-1} \cdot \sqrt{5}

  4. Substitute 1=i\sqrt{-1} = i: 5=i5\sqrt{-5} = i \cdot \sqrt{5}

So, the answer is: 5=i5\sqrt{-5} = i \sqrt{5}

Would you like a more detailed explanation, or do you have any questions?


Here are five related questions to deepen understanding:

  1. How would you express 10\sqrt{-10} as a complex number?
  2. What is the general form for the square root of any negative number in terms of ii?
  3. Can you simplify expressions involving higher powers of ii?
  4. What are some practical applications of imaginary numbers in physics or engineering?
  5. How do complex numbers help in solving quadratic equations with no real solutions?

Tip: When simplifying square roots of negative numbers, remember that ii allows you to express any negative square root in terms of positive values.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Imaginary Unit
Square Roots of Negative Numbers

Formulas

√a·b = √a · √b
√−1 = i

Theorems

Definition of the Imaginary Unit (i)

Suitable Grade Level

Grades 10-12