The expression 7+5i/1−4i in standard form is −13/17 + 33/17 i.
To write the expression (7 + 5i) / (1 - 4i) as a complex number in standard form, you can use the technique of multiplying the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.
The conjugate of 1 - 4i is 1 + 4i.
So, multiply both the numerator and denominator by the conjugate:
7+5i/ 1−4i ⋅ 1+4i /1+4i
Now, perform the multiplication:
Numerator:
(7+5i)(1+4i)=7+28i+5i+20i^2
=7+33i−20 (Remember that i^2 =−1)
=−13+33i
Denominator:
=(1−4i)(1+4i)=1+4i−4i−16i^2
=1- 6i^2
=1+16 (Again, i^2 =−1
=17
Now, the fraction becomes:
−13+33i/ 17
To express it in standard form, divide both the real and imaginary parts by 17:
−13/17 + 33/17 i.
So, the expression 7+5i/1−4i in standard form is −13/17 + 33/17 i.