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Find the distance d between z_(1)=(2+2i) and z_(2)=(6-4i). Express your answer in exact terms and simplify, if needed. d=

Find the distance d d between z1=(2+2i) z_{1}=(2+2 i) and z2=(6−4i) z_{2}=(6-4 i) . Express your answer in exact terms and simplify, if needed.\newlined= d=

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Q. Find the distance d d between z1=(2+2i) z_{1}=(2+2 i) and z2=(6−4i) z_{2}=(6-4 i) . Express your answer in exact terms and simplify, if needed.\newlined= d=
  1. Identify Formula: Identify the formula to calculate the distance between two complex numbers. The distance between two complex numbers z1z_1 and z2z_2 is given by the formula d=∣z2−z1∣d = |z_2 - z_1|, where ∣z∣|z| denotes the modulus of the complex number zz.
  2. Subtract Numbers: Subtract z1z_{1} from z2z_{2}. Given z1=2+2iz_{1} = 2 + 2i and z2=6−4iz_{2} = 6 - 4i, we calculate z2−z1=(6−4i)−(2+2i)=6−4i−2−2i=4−6iz_{2} - z_{1} = (6 - 4i) - (2 + 2i) = 6 - 4i - 2 - 2i = 4 - 6i.
  3. Calculate Modulus: Calculate the modulus of the difference.\newlineThe modulus of a complex number a+bia + bi is given by a2+b2\sqrt{a^2 + b^2}. So, for 4−6i4 - 6i, the modulus is 42+(−6)2=16+36=52\sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52}.
  4. Simplify Modulus: Simplify the modulus, if possible. 52\sqrt{52} can be simplified by factoring out perfect squares. 52=(4×13)=4×13=213\sqrt{52} = \sqrt{(4\times13)} = \sqrt{4} \times \sqrt{13} = 2\sqrt{13}.

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