Mathematics 10 — Exercise Worked examples · Example 5 (i)

Full question

Evaluate the expression.

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.

   $$-4\times i\times \left(5-6\times i\right)+3\times \left(5-6\times i\right)$$

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

   $$3\times 5+3\times \left(-1\right)\times 6\times i-4\times 5\times i-4\times \left(-6\right)\times ii$$

3. Real × real: multiply the numerical coefficients.

   $$3\times 5=15$$

4. Real × imaginary: multiply the numerical coefficients.

   $$3\times \left(-1\right)\times 6\times i=-18\times i$$

5. Imaginary × real: multiply the numerical coefficients.

   $$-4\times 5\times i=-20\times i$$

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$-4\times \left(-6\right)\times ii=24\times i^2$$

7. Substitute these four products into the expansion.

   $$15-18\times i-20\times i+24\times i^2$$

8. In the last term replace i² by −1.

   $$24\times i^2=24\times \left(-1\right)$$

9. Multiply its coefficient by −1.

   $$24\times \left(-1\right)=-24$$

10. Rewrite the expansion after this sign calculation.

    $$15-18\times i-20\times i-24$$

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

    $$15-24+i\times \left(-18-20\right)$$

12. Calculate the sum of the real terms.

    $$15-24=-9$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 20=-20$$

14. Calculate the sum of the imaginary coefficients.

    $$-18-20=-38$$

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

    $$-38\times i-9$$

Answer

The final answer is shown below.