Divide \$(5 + 7i)/(3 - 4i)\$ and simplify the result.

Given that

\$(5 + 7i)/(3 - 4i)\$

Identify the complex conjugate of the denominator:

The complex conjugate of (3 - 4i) is (3 + 4i)

Multiply both the numerator and denominator by the complex conjugate:

\$((5 + 7i)(3 + 4i)) / ((3 - 4i)(3 + 4i))\$

Expand the numerator:

\$ (5 \times 3) + (5 \times 4i) + (7i \times 3) + (7i \times 4i) \$

\$15 + 20i + 21i + 28i^2\$

15 + 41i - 28 (since \$i^2 = -1\$)

-13 + 41i

Expand the denominator:

\$(3 \times 3) + (3 \times 4i) + (-4i \times 3) + (-4i \times 4i)\$

\$9 + 12i - 12i - 16i^2\$

9 + 16 (since \$i^2 = -1\$)

25

Simplify the fraction:

\$(-13 + 41i) / 25\$

Separate into real and imaginary parts:

\$-13/25 + (41/25)i\$

⇒ -0.52 + 1.64i

Therefore, the simplified result of dividing (5 + 7i) by (3 - 4i) is -0.52 + 1.64i.

This option is incorrect. While the imaginary part matches our result, the real part should be negative, not positive

1 + 1.64i

This option is correct. It matches our calculated result of \$(5 + 7i)/(3 - 4i)\$ which is -0.52 + 1.64i

-0.52 + 1.64i

This option is incorrect. The real part should be negative and the imaginary part should be 1.64i, not 1.04i

1 + 1.04i

This option is incorrect. The real part is positive when it should be negative, and the imaginary part has the wrong sign

0.52 - 1.64i

Option

B

Can you summarize what you’ve understood in the above steps?