Q1. Write the number as the product of a real number and i:

Background

Topic: Complex Numbers

This question tests your understanding of how to express the square root of a negative number in terms of the imaginary unit .

Key Terms and Formulas:

• Imaginary unit: , where

• For any positive real number ,

Step-by-Step Guidance

1. Recognize that can be rewritten as .

2. Recall that .

3. Combine the results to write as a product of a real number and .

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Q2. Multiply or divide, as indicated. Simplify the answer:

Background

Topic: Multiplying Complex Numbers

This question tests your ability to multiply two imaginary numbers and simplify the result.

Key Terms and Formulas:

• Imaginary unit: , where

• for

Step-by-Step Guidance

1. Rewrite each as .

2. Multiply the two expressions: .

3. Use the property that and .

4. Combine the results to simplify the expression.

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Q3. Find the sum or difference. Write the answer in standard form:

Background

Topic: Addition of Complex Numbers

This question tests your ability to add complex numbers and write the result in standard form .

Key Terms and Formulas:

• Standard form: , where and are real numbers

• Add real parts together and imaginary parts together.

Step-by-Step Guidance

1. Identify the real and imaginary parts in each complex number.

2. Add the real parts: .

3. Add the imaginary parts: .

4. Combine the results to write in form.

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Q4. Simplify the power of :

Background

Topic: Powers of the Imaginary Unit

This question tests your understanding of the cyclical nature of powers of .

Key Terms and Formulas:

• (and the pattern repeats every 4 powers)

Step-by-Step Guidance

1. Divide the exponent (66) by 4 to find the remainder.

2. The remainder tells you which value in the cycle to use.

3. Express in terms of where is the remainder.

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Q5. Solve the quadratic equation:

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve a quadratic equation by factoring or using the quadratic formula.

Key Terms and Formulas:

• Quadratic equation:

• Factoring: Find two numbers that multiply to and add to .

• Quadratic formula:

Step-by-Step Guidance

1. Identify , , .

2. Look for two numbers that multiply to and add to $4$.

3. Write the equation in factored form if possible, or set up the quadratic formula.

4. Set each factor equal to zero and solve for .

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