Let z=-8+15i and w=6-8i. Compute
zz/ww
(each of the second ones, z and w, have a bar over them)
where the bar represents the complex conjugate.
recall that if
z = x+iy
zz* = (x+iy)(x-iy) = x^2+y^2
z = -8+15i and w = 6-8i. The equation would be ((-8 + 15i)(-8 - 15i))/((6 - 8i)(6+8i) Then you expand the binomials. That equals 64 + 120i - 120i - 225i^2. Then the two 120i's cancel to get 64 - 225i^2 for the numerator. Then you simplify the denominator the same way to get 36 + 24i - 24i - 64i^2. Again the 24i's cancel to get 36 - 64i^2. i^2 = -1, so (64 - 225(-1)) / (36 - 64(-1)) = (64 + 225) / (36 + 64). then you get the fraction 289/100.
289/100 is the answer
To compute zz/ww using the complex conjugate, we need to first find the values of zz and ww.
To calculate zz, we multiply z by its complex conjugate, which is z̄ (z with a bar over it):
zz = z * z̄
Similarly, to calculate ww, we multiply w with its complex conjugate, w̄:
ww = w * w̄
Step 1: Finding z̄
To find z̄, we need to take the complex conjugate of z. The complex conjugate of a complex number a + bi is simply a - bi, where a is the real part and b is the imaginary part.
Given z = -8 + 15i, the complex conjugate z̄ is -8 - 15i.
Step 2: Finding zz
Now we can calculate zz by multiplying z with its complex conjugate:
zz = (-8 + 15i) * (-8 - 15i)
To multiply complex numbers, we can use the distributive property. Multiply each term of the first complex number by each term of the second complex number and combine like terms:
zz = (-8 * -8) + (-8 * -15i) + (15i * -8) + (15i * -15i)
zz = 64 + 120i - 120i - 225i²
Remember that i² is equal to -1, so we can simplify further:
zz = 64 + 120i - 120i - 225(-1)
zz = 64 + 120i - 120i + 225
zz = 289
Step 3: Finding w̄
Now let's find w̄, the complex conjugate of w. Given w = 6 - 8i, w̄ is 6 + 8i.
Step 4: Finding ww
Next, we calculate ww by multiplying w with its complex conjugate:
ww = (6 - 8i) * (6 + 8i)
Using the distributive property, we can expand and combine like terms:
ww = (6 * 6) + (6 * 8i) + (-8i * 6) + (-8i * 8i)
ww = 36 + 48i - 48i - 64i²
Again, substitution i² with -1:
ww = 36 + 48i - 48i - 64(-1)
ww = 100
Step 5: Calculating zz/ww
Finally, we can compute zz/ww by dividing zz by ww:
zz/ww = 289/100
Therefore, zz/ww equals 2.89.
So, the resulting value of zz/ww, using complex conjugates, is 2.89.