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Find the distance d between z_(1)=(8+3i) and z_(2)=(5-7i). Express your answer in exact terms and simplify, if needed. d=

Find the distance d d between z1=(8+3i) z_{1}=(8+3 i) and z2=(5−7i) z_{2}=(5-7 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=(8+3i) z_{1}=(8+3 i) and z2=(5−7i) z_{2}=(5-7 i) . Express your answer in exact terms and simplify, if needed.\newlined= d=
  1. Identify formula for distance: Identify the formula for the distance between two complex numbers. The distance between two complex numbers z1=(x1+yi1)z_1 = (x_1 + yi_1) and z2=(x2+yi2)z_2 = (x_2 + yi_2) is given by the formula d=((x2−x1)2+(y2−y1)2)d = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}.
  2. Plug values into formula: Plug the values of z1z_1 and z2z_2 into the distance formula.\newlineFor z1=(8+3i)z_1 = (8 + 3i) and z2=(5−7i)z_2 = (5 - 7i), we have x1=8x_1 = 8, y1=3y_1 = 3, x2=5x_2 = 5, and y2=−7y_2 = -7. So, we substitute these values into the formula to get d=((5−8)2+(−7−3)2)d = \sqrt{((5 - 8)^2 + (-7 - 3)^2)}.
  3. Calculate differences: Calculate the differences (x2−x1)(x_2 - x_1) and (y2−y1)(y_2 - y_1). The differences are (5−8)=−3(5 - 8) = -3 and (−7−3)=−10(-7 - 3) = -10.
  4. Square differences and add: Square the differences and add them. Squaring the differences gives us (−3)2=9(-3)^2 = 9 and (−10)2=100(-10)^2 = 100. Adding them together yields 9+1009 + 100.
  5. Calculate square root of sum: Calculate the square root of the sum to find the distance.\newlineThe sum is 9+100=1099 + 100 = 109. Taking the square root gives us d=109d = \sqrt{109}.

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