Question

question 10
simplify.
(6 + 7i)(3 - 4i)
10 - 3i
46 + 3i
10 + 3i
46 - 3i

Explanation:

Step1: Apply distributive property (FOIL)

Multiply each term in the first complex number by each term in the second:
$(6 + 7i)(3 - 4i) = 6 \cdot 3 + 6 \cdot (-4i) + 7i \cdot 3 + 7i \cdot (-4i)$

Step2: Simplify each product

Calculate each term:
$6 \cdot 3 = 18$,
$6 \cdot (-4i) = -24i$,
$7i \cdot 3 = 21i$,
$7i \cdot (-4i) = -28i^2$.

Step3: Replace \(i^2\) with \(-1\)

Since \(i^2 = -1\), substitute:
$-28i^2 = -28(-1) = 28$.

Step4: Combine like terms

Combine real parts and imaginary parts:
Real: \(18 + 28 = 46\),
Imaginary: \(-24i + 21i = -3i\)? Wait, no—wait, recheck:
Wait, \(-24i + 21i = -3i\)? Wait, no, wait: Wait, \(6 \cdot 3 = 18\), \(7i \cdot (-4i) = -28i^2 = 28\), so real parts: \(18 + 28 = 46\). Imaginary parts: \(6 \cdot (-4i) + 7i \cdot 3 = -24i + 21i = -3i\)? Wait, but the options have \(46 - 3i\)? Wait, let's recalculate:

Wait, \((6 + 7i)(3 - 4i) = 63 + 6(-4i) + 7i3 + 7i(-4i)\)
= \(18 -24i +21i -28i^2\)
= \(18 -3i -28(-1)\) (since \(i^2 = -1\))
= \(18 -3i +28\)
= \(46 -3i\).

Wait, but the options include \(46 - 3i\) as the last option. Wait, but let me check again. Wait, maybe I made a sign error. Wait, \(7i*3 = 21i\), \(6*(-4i) = -24i\), so \(21i -24i = -3i\). Then \(18 +28 = 46\), so total is \(46 -3i\). Wait, but the options: the last option is \(46 - 3i\). Wait, but let me check the original problem again. Wait, the options are:

1. \(10 - 3i\)
2. \(46 + 3i\)
3. \(10 + 3i\)
4. \(46 - 3i\)

Ah, so the correct calculation gives \(46 - 3i\), which is the fourth option. Wait, but let me re-express the steps:

Wait, \((a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\) (since \(i^2 = -1\)). Let's use this formula:

Here, \(a = 6\), \(b = 7\), \(c = 3\), \(d = -4\).

So real part: \(ac - bd = 63 - (7)(-4) = 18 +28 = 46\).

Imaginary part: \(ad + bc = 6(-4) + 73 = -24 +21 = -3\). So the imaginary part is \(-3i\). Thus, the result is \(46 -3i\), which is the fourth option.

Answer:

D. \(46 - 3i\) (assuming the options are labeled A to D as: A. \(10 - 3i\), B. \(46 + 3i\), C. \(10 + 3i\), D. \(46 - 3i\))