Prove that the value of the determinant `|-7 5+3i2/3-4i5-3i8 4+5i2/3+4i4-fi9|` is real.

Prove that the value of the determinant |{:(-7,,5+3i,,(2)/(3)-4i),(5-3i ,,8,,4+5i),((2)/(3) +4i,,4-5i,,9):}|" is real "

Find the value of : i^4 + i^5 + i^6 + i^7

Find the value of : i^4 + i^5 + i^6 + i^7

Find the value of i^4 + i^5 + i^6 + i^7

Whitout expanding the determinat at any stage prove that |{:(-5,3+5i,(3)/(2)-4i),(3-5i,8,4+5i),((3)/(2)+4i,4-5i,9):}| has a purely real value.

Prove that the points representing the complex numbers 4+3i, 6+4i, 5+6i,3+5i are the vertices of a square.

Prove that the following complex numbers are purely real: (i) ((2+3i)/(3+4i))((2-3i)/(3-4i)) (ii) ((3+2i)/(2-3i))((3-2i)/(2+3i))

Show that the complex number ((4+3i)/(3 + 4i)) ((4 -3i)/(3-4i)) is purely real.

Express the following in the form of a+ib (2+3i)/(5-4i)+(2-3i)/(5+4i)