If (x+iy)31=a+ib, where x,y,a,b∈R, show that ax−by=−2(a2+b2)
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Step-by-Step Solution
Step 1: Cube both sides of the equation
To eliminate the fractional exponent, we cube both sides of the given equation. This allows us to express x+iy in terms of a and b.
Step 2: Expand the right side using the binomial theorem
We expand the term (a+ib)3 using the binomial theorem (A+B)3=A3+3A2B+3AB2+B3. Remember that i2=−1 and i3=−i. We then group the real and imaginary parts.
Step 3: Equate real and imaginary parts
Since x+iy=(a3−3ab2)+i(3a2b−b3), and x, y, a, b are real numbers, we can equate the real parts and the imaginary parts from both sides of the equation.
Step 4: Simplify the terms ax and by
We divide the expression for x by a and the expression for y by b. We can factor out a from the expression for x and b from the expression for y before dividing, assuming a=0 and b=0.
Step 5: Substitute and simplify to show the required identity
Finally, we substitute the simplified expressions for ax and by into the left side of the identity we need to prove. Combining like terms leads directly to the right side of the identity.