Math Problem Statement

Use the imaginary number 𝑖 to rewrite the expression below as a complex number. Simplify all radicals. – – 25

Solution

To rewrite the expression βˆ’βˆ’25-\sqrt{-25} as a complex number, we can use the imaginary unit ii, where i=βˆ’1i = \sqrt{-1}.

  1. Start with the expression: βˆ’βˆ’25-\sqrt{-25}

  2. Rewrite βˆ’25\sqrt{-25} using the imaginary unit ii: βˆ’25=25β‹…(βˆ’1)=25β‹…βˆ’1=5i\sqrt{-25} = \sqrt{25 \cdot (-1)} = \sqrt{25} \cdot \sqrt{-1} = 5i

  3. Now, include the negative sign from the original expression: βˆ’βˆ’25=βˆ’5i-\sqrt{-25} = -5i

Thus, the expression βˆ’βˆ’25-\sqrt{-25} rewritten as a complex number is: βˆ’5i-5i

Would you like more details or have any questions about this? Here are some related questions to explore further:

  1. What is the value of βˆ’9\sqrt{-9} in terms of ii?
  2. How would you express βˆ’36\sqrt{-36} as a complex number?
  3. If z=4βˆ’3iz = 4 - 3i, what is the magnitude of zz?
  4. How do you multiply two complex numbers, say 3+4i3 + 4i and 2βˆ’5i2 - 5i?
  5. What is the complex conjugate of 7+2i7 + 2i?

Tip: Remember that i2=βˆ’1i^2 = -1. This is fundamental when simplifying expressions involving imaginary numbers.

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Math Problem Analysis

Mathematical Concepts

Imaginary Numbers
Complex Numbers
Radicals

Formulas

i = √(-1)
√(a * b) = √a * √b

Theorems

Properties of Square Roots with Negative Numbers

Suitable Grade Level

Grades 9-11