Question

Given z equals quantity negative 11 minus 4 times i end quantity over quantity 3 plus 5 times i end quantity comma rewrite z in standard form.

•24/16 + 19/16 ;
•17/20 + 19/20 ;

Answer 1

To rewrite z in standard form, we can multiply the numerator and denominator by the conjugate of the denominator to eliminate the 'i' terms in the denominator. The conjugate of 3 + 5i is 3 - 5i. After simplifying the expression, the standard form of z is -13/34 + 67/34i.

Step-by-step explanation:

To rewrite z in standard form, we need to separate the real part (the part without 'i') from the imaginary part (the part with 'i'). In the given expression, z = (-11 - 4i) / (3 + 5i). To get rid of the denominator, we can multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 3 + 5i is 3 - 5i. Multiplying the numerator and denominator by this conjugate will eliminate the 'i' terms in the denominator.

Multiplying: z = [(-11 - 4i) / (3 + 5i)] * [(3 - 5i) / (3 - 5i)]

Expanding and simplifying: z = [(-33 + 55i + 12i - 20i^2) / (9 - 25i^2)]

Simplifying further: z = [(-33 + 67i - 20i^2) / (9 + 25)]

Since i^2 is equal to -1, we have: z = (-33 + 67i + 20) / 34

Simplifying the expression: z = (−13 + 67i) / 34. Thus, the standard form of z is -13/34 + 67/34i.

Answer 2

To rewrite z in standard form, we need to multiply both the numerator and denominator by the conjugate of the denominator. Simplifying the expression, we get -53/34 + 43/34i.

The standard form of a complex number is a+bi,

where a and b are real numbers.

The given complex number is z = -11 - 4i / 3 + 5i.

To rewrite z in standard form, we need to multiply both the numerator and denominator by the conjugate of the denominator.

The conjugate of 3 + 5i is 3 - 5i.

Multiplying the numerator and denominator by 3 - 5i,

z = ( -11 -4i )( 3 - 5i ) / ( 3 + 5i )( 3 - 5i )

z = (-33 + 55i - 12i + 20i^2) / (9 - 25i^2).

Combining like terms and simplifying,

z = (-53 + 43i) / 34.

Dividing both the real and imaginary parts by 34,

z = -53/34 + 43/34i.

Therefore, the standard form of the complex number z is -53/34 + 43/34i.