Math Problem Statement

Solve

Solution

The question asks for the square root of two values:

  1. i\sqrt{i}
  2. 8\sqrt{-8}

Part (a): i\sqrt{i}

The square root of the imaginary unit ii can be determined using the following method:

  1. Express ii in polar form: i=eiπ2i = e^{i\frac{\pi}{2}}.
  2. Take the square root: i=eiπ2=eiπ4\sqrt{i} = \sqrt{e^{i\frac{\pi}{2}}} = e^{i\frac{\pi}{4}}.
  3. Convert back to rectangular form: i=cos(π4)+isin(π4)=12+i12\sqrt{i} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} This simplifies to: i=22+i22\sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}

Part (b): 8\sqrt{-8}

To find the square root of 8-8, we can express 8-8 as 8×(1)8 \times (-1):

  1. Separate the negative part: 8=8×1\sqrt{-8} = \sqrt{8} \times \sqrt{-1}.
  2. Simplify: 8=22\sqrt{8} = 2\sqrt{2} and 1=i\sqrt{-1} = i.
  3. Combine: 8=22i\sqrt{-8} = 2\sqrt{2}i

Final Answers:

  • i=22+i22\sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}
  • 8=22i\sqrt{-8} = 2\sqrt{2}i

Would you like more details or have any questions?

Here are some related questions for further practice:

  1. What is the square root of 4-4?
  2. How do you express ii in exponential form?
  3. What is the square root of 8i8i?
  4. How can we represent complex numbers on an Argand diagram?
  5. What is the polar form of the complex number i-i?
  6. How do you multiply two complex numbers in polar form?
  7. What are the roots of the equation x2+1=0x^2 + 1 = 0?
  8. How do you add and subtract complex numbers?

Tip: When dealing with square roots of negative numbers, it's often useful to convert the negative number into a form that includes ii to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Imaginary Unit
Square Roots

Formulas

Polar form of complex numbers
Square root of a complex number

Theorems

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Suitable Grade Level

Advanced High School and Beyond