Question

simplify. (example 3)

1. (-2 - i)^2
2. (5 + 2i)^2
3. (2 + i)(4 + 3i)
4. (5 + 3i)(2 + 6i)
5. (1 + 4i)^2
6. (3 + i)^2
7. (3 + 5i)(3 - 5i)
8. (6 + 7i)(6 - 7i)

Answer:

1. For problem 18: Simplify \((-2 - i)^2\)

• Step 1: Apply the formula \((a + b)^2=a^{2}+2ab + b^{2}\)
• Here \(a=-2\) and \(b = -i\). So \((-2 - i)^2=(-2)^2+2\times(-2)\times(-i)+(-i)^2\).
• Calculate each term: \((-2)^2 = 4\), \(2\times(-2)\times(-i)=4i\), and \((-i)^2=- 1\) (since \(i^2=-1\)).
• Then \((-2 - i)^2=4 + 4i-1\).
• Step 2: Combine like - terms
• Combine the real parts: \(4-1 = 3\). So \((-2 - i)^2=3 + 4i\).

1. For problem 19: Simplify \((1 + 4i)^2\)

• Step 1: Apply the formula \((a + b)^2=a^{2}+2ab + b^{2}\)
• Here \(a = 1\) and \(b = 4i\). So \((1 + 4i)^2=1^{2}+2\times1\times4i+(4i)^2\).
• Calculate each term: \(1^{2}=1\), \(2\times1\times4i = 8i\), and \((4i)^2=16i^{2}\). Since \(i^{2}=-1\), \((4i)^2=-16\).
• Then \((1 + 4i)^2=1 + 8i-16\).
• Step 2: Combine like - terms
• Combine the real parts: \(1-16=-15\). So \((1 + 4i)^2=-15 + 8i\).

1. For problem 20: Simplify \((5 + 2i)^2\)

• Step 1: Apply the formula \((a + b)^2=a^{2}+2ab + b^{2}\)
• Here \(a = 5\) and \(b = 2i\). So \((5 + 2i)^2=5^{2}+2\times5\times2i+(2i)^2\).
• Calculate each term: \(5^{2}=25\), \(2\times5\times2i = 20i\), and \((2i)^2 = 4i^{2}\). Since \(i^{2}=-1\), \((2i)^2=-4\).
• Then \((5 + 2i)^2=25+20i - 4\).
• Step 2: Combine like - terms
• Combine the real parts: \(25 - 4=21\). So \((5 + 2i)^2=21 + 20i\).

1. For problem 21: Simplify \((3 + i)^2\)

• Step 1: Apply the formula \((a + b)^2=a^{2}+2ab + b^{2}\)
• Here \(a = 3\) and \(b = i\). So \((3 + i)^2=3^{2}+2\times3\times i+i^{2}\).
• Calculate each term: \(3^{2}=9\), \(2\times3\times i = 6i\), and \(i^{2}=-1\).
• Then \((3 + i)^2=9+6i - 1\).
• Step 2: Combine like - terms
• Combine the real parts: \(9 - 1=8\). So \((3 + i)^2=8 + 6i\).

1. For problem 22: Simplify \((2 + i)(4 + 3i)\)

• Step 1: Use the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\)
• Here \(a = 2\), \(b = i\), \(c = 4\), and \(d = 3i\). So \((2 + i)(4 + 3i)=2\times4+2\times3i+i\times4+i\times3i\).
• Calculate each term: \(2\times4 = 8\), \(2\times3i=6i\), \(i\times4 = 4i\), and \(i\times3i = 3i^{2}\). Since \(i^{2}=-1\), \(i\times3i=-3\).
• Then \((2 + i)(4 + 3i)=8+6i + 4i-3\).
• Step 2: Combine like - terms
• Combine the real parts: \(8-3 = 5\), and combine the imaginary parts: \(6i+4i = 10i\). So \((2 + i)(4 + 3i)=5 + 10i\).

1. For problem 23: Simplify \((3 + 5i)(3 - 5i)\)

• Step 1: Use the difference - of - squares formula \((a + b)(a - b)=a^{2}-b^{2}\)
• Here \(a = 3\) and \(b = 5i\). So \((3 + 5i)(3 - 5i)=3^{2}-(5i)^{2}\).
• Calculate each term: \(3^{2}=9\), \((5i)^{2}=25i^{2}\). Since \(i^{2}=-1\), \((5i)^{2}=-25\).
• Then \((3 + 5i)(3 - 5i)=9-(-25)\).
• Step 2: Simplify the expression
• \(9-(-25)=9 + 25=34\).

1. For problem 24: Simplify \((5 + 3i)(2 + 6i)\)

• Step 1: Use the FOIL method \((a + b)(c + d)=ac+ad+bc+bd\)
• Here \(a = 5\), \(b = 3i\), \(c = 2\), and \(d = 6i\). So \((5 + 3i)(2 + 6i)=5\times2+5\times6i+3i\times2+3i\times6i\).
• Calculate each term: \(5\times2 = 10\), \(5\times6i=30i\), \(3i\times2 = 6i\), and \(3i\times6i = 18i^{2}\). Since \(i^{2}=-1\), \(3i\times6i=-18\).
• Then \((5 + 3i)(2 + 6i)=10+30i + 6i-18\).
• Step 2: Combine like - terms
• Combine the real parts: \(10-18=-8\), and combine the imaginary parts: \(30i+6i = 36i\). So \((5 + 3i)(2 + 6i)=-8 + 36i\).

1. For problem 25: Simplify \((6 + 7i)(6 - 7i)\)

• Step 1: Use the difference - of - squares formula \((a + b)(a - b)=a^{2}-b^{2}\)
• Here \(a = 6\) and \(b = 7i\). So \((6 + 7i)(6 - 7i)=6^{2}-(7i)^{2}\).
• Calculate each term: \(6^{2}=36\), \((7i)^{2}=49i^{2}\). Since \(i^{2}=-1\), \((7i)^{2}=-49\).
• Then \((6 + 7i)(6 - 7i)=36-(-49)\).
• Step 2: Simplify the expression
• \(36-(-49)=36 + 49 = 85\).

1. # Answer: \(3 + 4i\)
2. # Answer: \(-15 + 8i\)
3. # Answer: \(21 + 20i\)
4. # Answer: \(8 + 6i\)
5. # Answer: \(5 + 10i\)
6. # Answer: \(34\)
7. # Answer: \(-8 + 36i\)
8. # Answer: \(85\)