Math Problem Statement

Simplify the expression to a, plus, b, ia+bi form:

minus, square root of, 36, end square root, minus, square root of, minus, 9, end square root, minus, square root of, 1, end square root, minus, square root of, minus, 64, end square root

Solution

Let's simplify the given expression step by step.

The expression is:

369164-\sqrt{36} - \sqrt{-9} - \sqrt{1} - \sqrt{-64}

Step 1: Simplify each square root term.

  1. 36=6\sqrt{36} = 6, so 36=6-\sqrt{36} = -6.
  2. 9\sqrt{-9} involves the square root of a negative number, which can be written as 9=9×1=3i\sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i, so 9=3i-\sqrt{-9} = -3i.
  3. 1=1\sqrt{1} = 1, so 1=1-\sqrt{1} = -1.
  4. 64\sqrt{-64} can be written as 64×1=8i\sqrt{64} \times \sqrt{-1} = 8i, so 64=8i-\sqrt{-64} = -8i.

Step 2: Combine the simplified terms.

Now, the expression becomes:

63i18i-6 - 3i - 1 - 8i

Step 3: Group the real and imaginary terms.

Combine the real parts and the imaginary parts:

(61)+(3i8i)=711i(-6 - 1) + (-3i - 8i) = -7 - 11i

Final Answer:

The expression simplified to a+bia + bi form is:

711i\boxed{-7 - 11i}

Would you like details on any step or have any further questions? Here are some related questions:

  1. How do you simplify complex numbers involving square roots of negative numbers?
  2. What is the significance of the imaginary unit ii in mathematics?
  3. How do you add or subtract complex numbers?
  4. What are some real-life applications of complex numbers?
  5. How do you multiply and divide complex numbers?

Tip: Always remember that the square root of a negative number introduces the imaginary unit ii, where i=1i = \sqrt{-1}.

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Math Problem Analysis

Mathematical Concepts

Square Roots
Complex Numbers

Formulas

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Theorems

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Suitable Grade Level

Advanced High School