Mathematics 10 — Exercise Skilled practice · Page 17 n=2

Full question

Simplify the following expression and find its real and imaginary parts.

Step-by-step solution

1. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

   $$i\times y\times \left(i\times y+x\right)+x\times \left(i\times y+x\right)$$

2. Distribute each of those products. Keep all four multiplications visible.

   $$i\times x\times y+i\times x\times y+i\times i\times yy+xx$$

3. Real × real: multiply the numerical coefficients.

   $$xx=x^2$$

4. Real × imaginary: multiply the numerical coefficients.

   $$i\times x\times y=i\times \left(x\times y\right)$$

5. Imaginary × real: multiply the numerical coefficients.

   $$i\times x\times y=i\times \left(x\times y\right)$$

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$i\times i\times yy=i^2\times y^2$$

7. Substitute these four products into the expansion.

   $$x^2+i\times \left(x\times y\right)+i\times \left(x\times y\right)+i^2\times y^2$$

8. In the last term replace i² by −1.

   $$i^2\times y^2=\left(-1\right)\times y^2$$

9. Multiply its coefficient by −1.

   $$\left(-1\right)\times y^2=-y^2$$

10. Rewrite the expansion after this sign calculation.

    $$x^2+i\times \left(x\times y\right)+i\times \left(x\times y\right)-y^2$$

11. Group the real terms and factor i out of the imaginary terms.

    $$x^2-y^2+i\times \left(x\times y+x\times y\right)$$

12. Calculate the sum of the real terms.

    $$x^2-y^2=x^2-y^2$$

13. Calculate the sum of the imaginary coefficients.

    $$x\times y+x\times y=2\times x\times y$$

14. Insert both calculated sums in a + bi form.

    $$x^2-y^2+i\times \left(2\times x\times y\right)$$

15. Now read the constant term as Re(z) and the coefficient of i as Im(z).

    $$(\mathrm{Re}=x^2-y^2,\mathrm{Im}=2\times x\times y)$$

Answer

The final answer is shown below.