Question:

Find the multiplicative inverse of the complex number −i

Answer:

Answer: Step 1: The multiplicative inverse of any number is the reciprocal of the number.

Step 2: The reciprocal of -i is 1/-i.

Step 3: Therefore, the multiplicative inverse of -i is 1/-i.

Question:

Express the given complex number in the form a+ib:(1−i)⁴

Answer:

(1-i)^4 = (1-i)(1-i)(1-i)(1-i) = (1-i)(1-i)(1-i-i) = (1-i)(1-2i) = (1-2i-i^2) = (1-2i-1) = 1-2i-1+i^2 = 1-2i-1+(-1) = 1-2i-2 = 1-2(i+1) = 1-2i-2i = 1-4i = a+ib = 1-4i

Therefore, the complex number in the form a+ib is 1-4i.

Question:

Find the multiplicative inverse of the complex number √5+3i.

Answer:

Answer: Step 1: Find the conjugate of the complex number √5+3i. Conjugate = √5-3i

Step 2: Find the modulus of the complex number √5+3i. Modulus = |√5+3i| = √(5^2 + 3^2) = √34

Step 3: Find the multiplicative inverse of the complex number √5+3i. Multiplicative inverse = (√5-3i)/√34

Question:

(5i)(−3​/5i)

Answer:

1. First, combine the two terms by adding them: 5i + (-3/5i)

2. Simplify the terms by multiplying the numerator and denominator of the second term by the complex conjugate of the denominator of the second term: 5i + (-3/(5i)(-5i)) = 5i + (15/25)

3. Simplify the terms by combining the coefficients of the terms: 5i + 15/25 = (5i + 15)/25

4. Simplify the terms by multiplying the numerator and denominator by the reciprocal of the denominator: (5i + 15)/25 * 25/25 = (5i + 15)/25 * 1 = (5i + 15)

Question:

Solve : i⁹+i¹⁹

Answer:

Step 1: Rewrite the expression using the power of a power rule.

i^9 + i^19 = i^(9+19)

Step 2: Calculate the value of i^(9+19).

i^(9+19) = i^28

Question:

Express the given complex number 3(7+i7)+i(7+i7) in the form a+ib,

Answer:

Given: 3(7+i7)+i(7+i7)

Step 1: 3(7+i7) = 21 + 3i7

Step 2: i(7+i7) = -7 + i7

Step 3: 21 + 3i7 + (-7 + i7)

Step 4: 21 - 7 + 4i7

Answer: 21 - 7 + 4i7 = 14 + 4i7

Question:

Express the given complex number in the form a+ib: (−2−1​/3i)³

Answer:

Given complex number, z = (-2 - i/3)

Step 1: Calculate z^3

z^3 = (-2 - i/3)^3

Step 2: Expand the expression

(-2 - i/3)^3 = -8 - 8i/3 - 2i^2/3 - (i/3)^3

Step 3: Simplify the expression

-8 - 8i/3 - 2i^2/3 - (i/3)^3 = -8 - 8i/3 - 2i^2/3 - i/27

Step 4: Express in the form a+ib

-8 - 8i/3 - 2i^2/3 - i/27 = -8 - 8i/3 - 2i^2/3 - i/27 = -8 - 8i/3 - 2i - i/9 = -8 - 10i - i/9

Therefore, the given complex number in the form a+ib is: -8 - 10i - i/9

Question:

Express the following expression in the form of a+ib ; (3+i√5)(3−i√5)/(√3+√2i)−(√3−i√2)

Answer:

Given, (3+i√5)(3−i√5)/(√3+√2i)−(√3−i√2)

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

(3+i√5)(3−i√5)(√3-√2i)/((√3+√2i)(√3-√2i))-((√3−i√2)(√3-√2i))

Step 2: Simplify the numerator and denominator.

(3+i√5)(3−i√5)(3+√2i)/(9+2i√6)-(3+√2i)(3+√2i)

Step 3: Simplify the numerator and denominator further.

(9-5√5)/(9+2i√6)-9

Step 4: Simplify the expression further.

-14-2i√6/9+2i√6

Step 5: Express the expression in the form of a+ib.

-14/9+2i√6/9

Question:

Solve: [(1​/3+i7​/3)+(4+i1​/3)]−(−4​/3+i)

Answer:

Step 1: Combine like terms on the left side of the equation: [(1​/3+i7​/3)+(4+i1​/3)]−(−4​/3+i)

Step 2: [(5​/3+i8​/3)]−(−4​/3+i)

Step 3: [5​/3+i8​/3 + 4​/3-i]

Step 4: 9​/3+i7​/3

Question:

Express (31​+3i)3 in the form a+ib. A 343​/34+23i B −343​/34−23i C 242​/27+26i D −242​/27−26i

Answer:

Answer: C 242​/27+26i

Question:

Find the modulus and amplitude of 4+3i.

Answer:

Answer: Modulus: 5 Amplitude: 5

Question:

Express (1−i)−(1+i6) as a+ib

Answer:

(1−i)−(1+i6) = (1−i) − 1 - i6 = -i - 1 - i6 = -i - 1 - (i/6) = -i - 1 - (1/6)i = -i - 1 + (-1/6)i = -i - 1 + (1/6)i = (-1 - 1 + (1/6))i = (-2 + (1/6))i = (-2 + (1/6))(1 + i) = (-2 + (1/6)) + (-2 + (1/6))i = (-2 + (1/6)) + i(-2 + (1/6)) = a + ib = (-2 + (1/6)) + i(-2 + (1/6)) = (-2 + (1/6)) + i(-2 + (1/6))

Question:

Solve the problem:- (1​/5+i2​/5)−(4+i5​/2)

Answer:

Given, (1/5 + i2/5) - (4+i5/2)

Step 1: Rewrite the given expression as,

(1/5 + i2/5) - (4+ i5/2) = (1/5 - 4) + (i2/5 - i5/2)

Step 2: Simplify the expression,

(1/5 - 4) + (i2/5 - i5/2) = -3.8 + (-3.5i)

Hence, the solution is -3.8 + (-3.5i).

Question:

Evaluate i⁻³⁹.

Answer:

i^−39 = 1/i^39

= 1/(i^2)^19

= 1/i^38

= 1/i^2 * 1/i^36

= 1/i^2 * (1/i^18)^2

= 1/i^2 * (1/i^2)^18

= (1/i^2)^19