Simplify (3-4i)(5-8i)

The sum of a real number and an imaginary number is termed a Complex number. These are the numbers that can be written in the form of a+ib, where a and b both are real numbers. It is denoted by z

Here in complex number form the value ‘a’ is called the real part which is denoted by Re(z), and ‘b’ is called the imaginary part Im(z). It is also called an imaginary number. In complex numbers form a +bi, ‘i’ is an imaginary number called “iota”.

The value of i is (√-1) or we can write as i² = -1.

For example:

• 3+4i is a complex number, where 3 is a real number (Re) and 4i is an imaginary number (Im).
• 2+5i is a complex number where 2 is a real number (Re) and  5i is an imaginary number (im)

The Combination of real number and imaginary number is called a Complex number.

Real numbers

The number represent in a number system such as positive, negative, zero, integer, rational, irrational, fractions, etc. are called real numbers. Real numbers are represented as Re().  

For example: 13, -45, 0, 1/7, 2.8, √5, etc., are all real numbers.

Imaginary numbers

The numbers which are not real are termed Imaginary numbers. After squaring an imaginary number, it gives a result in negative. Imaginary numbers are represented as Im().    

Example: √-3, √-7, √-11 are all imaginary numbers. here ‘i’ is an imaginary number called “iota”.

Rules of imaginary numbers:

i = √-1

i² = -1

i³ = -i

i⁴ = 1

i⁴ⁿ = 1

i^4n-1 = -1

Simplify (3-4i)(5-8i)

Solution: 

Given: (3-4i)(5-8i) 

= {15 – 24i – 20i + 32i²}

= {15 -24i – 20i + 32(-1)}

= {15 – 44i -32}

= -17 -44i

Similar Questions

Question 1: What is the answer to the following problem? (-3i)(7i)(-1)

Solution: 

Given : (-3i)(7i)(-1)

= -3i x 7i x (-1)

=  -21i² x -1                            { i² = -1}

= -21 (-1) x -1

= 21 x -1

= -21 + 0i

Question 2: Simplify: (2-4i)(5-7i).

Solution: 

Given :  (2-4i)(5-7i)

 =  10 -14i -20i +28i²

 =  10 -14i -20i + 28(-1)²

 = 10 – 14i – 20i +28

 =   18 – 34i

Question 3: Express the problem in a+ib, (-5i)(1/7i)

Solution:

Given : (-5i)(1/7i)

= -5/7 i²

= -5/7 (-1)                            { i² = -1}

=  5/7 + 0i

Question 4: Express in form of a+ib, {3(4+7i) + i(7+7i)}?

Solution:

Given : {3(4+7i) + i(7+7i) }

=    { 12 +21i + 7i + 7i² }

=    { 12 + 28i + 7(-1) }

=     { 12 + 28i – 7 }

=     15 + 28i

=   5 + 28i

Question 5: Simplify (3 + 4i) / (3 + 2i)

Solution:

Multiplying the numerator and denominator with the conjugate of denominators.

= ((3 + 4i) * (3 – 2i)) / ((3 + 2i) * (3 – 2i))

=(9 -6i +12i – 8i² ) / {9  -(2i)² }

=(17 + 6i) / (13)

=(17+ 6i) / 13

Question 6: Simplify in form of a + ib, (-5i)(1/8i)

Solution:

Given : (-5i)(1/8i)

     = (- 5/8 )i²

     =  (- 5/8 )(-1)

     = 5/8 + 0i

Question 7: Express in a+ib, {1/(3-4i)}

Solution: 

Given :  {1/(3-4i) }

      =  {1/(3-4i) } x {(3+ 4i)(3+ 4i)}

     =  (3+ 4i) / { 9 – 16i² }

     = (3+ 4i) / ( 9 +16)

    = (3+ 4i)/25

    = 3/25 + 4i /25