Q1. Simplify: (7 - i)(2 + 3i)

Background

Topic: Complex Numbers – Multiplication

This question tests your ability to multiply two complex numbers and express the result in standard form (a + bi).

Key Terms and Formulas

• Complex number: , where and are real numbers, and is the imaginary unit ().

• Multiplying complex numbers: Use the distributive property (FOIL method).

Step-by-Step Guidance

1. Expand using the distributive property (FOIL).

2. Multiply each term: , , , .

3. Combine like terms and remember that .

4. Group the real and imaginary parts together.

Try solving on your own before revealing the answer!

Q2. Simplify: (5 + 4i)(5 - 4i)

Background

Topic: Complex Numbers – Multiplication (Conjugates)

This question tests your understanding of multiplying a complex number by its conjugate, which results in a real number.

Key Terms and Formulas

• Conjugate: For , the conjugate is .

• Product of conjugates: (since ).

Step-by-Step Guidance

1. Expand using the distributive property.

2. Multiply: , , , .

3. Combine like terms and simplify .

4. Notice that the imaginary parts cancel out, leaving a real number.

Try solving on your own before revealing the answer!

Q3. Simplify:

Background

Topic: Complex Numbers – Powers and Square Roots

This question tests your ability to work with square roots of negative numbers and expand binomials involving complex numbers.

Key Terms and Formulas

• (since ).

• Binomial square: .

Step-by-Step Guidance

1. Rewrite as .

2. Expand using the binomial square formula.

3. Calculate each term: , , .

4. Simplify and combine like terms.

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Q4. Simplify:

Background

Topic: Complex Numbers – Division

This question tests your ability to divide complex numbers by multiplying numerator and denominator by the conjugate of the denominator.

Key Terms and Formulas

• Conjugate: For , the conjugate is .

• To divide: Multiply numerator and denominator by the conjugate of the denominator.

Step-by-Step Guidance

1. Identify the conjugate of the denominator: .

2. Multiply numerator and denominator by .

3. Expand both numerator and denominator using distributive property.

4. Simplify and combine like terms.

Try solving on your own before revealing the answer!

Q5. Simplify:

Background

Topic: Complex Numbers – Square Roots

This question tests your ability to simplify square roots of negative numbers and combine them.

Key Terms and Formulas

• Simplify each radical and combine like terms.

Step-by-Step Guidance

1. Rewrite as and as .

2. Simplify and into their simplest radical form.

3. Combine the terms, factoring out if possible.

Try solving on your own before revealing the answer!