Mathematics 10 — Exercise 1.2 · Q1 (vi)

Full question

Simplify the following expression and express the result in the standard form a + bi, where a and b are real numbers.

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.

   $$-2\times i\times \left(3-4\times i\right)+5\times \left(3-4\times i\right)$$

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

   $$5\times 3+5\times \left(-1\right)\times 4\times i-2\times 3\times i-2\times \left(-4\right)\times ii$$

3. Real × real: multiply the numerical coefficients.

   $$5\times 3=15$$

4. Real × imaginary: multiply the numerical coefficients.

   $$5\times \left(-1\right)\times 4\times i=-20\times i$$

5. Imaginary × real: multiply the numerical coefficients.

   $$-2\times 3\times i=-6\times i$$

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$-2\times \left(-4\right)\times ii=8\times i^2$$

7. Substitute these four products into the expansion.

   $$15-20\times i-6\times i+8\times i^2$$

8. In the last term replace i² by −1.

   $$8\times i^2=8\times \left(-1\right)$$

9. Multiply its coefficient by −1.

   $$8\times \left(-1\right)=-8$$

10. Rewrite the expansion after this sign calculation.

    $$15-20\times i-6\times i-8$$

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

    $$15-8+i\times \left(-20-6\right)$$

12. Calculate the sum of the real terms.

    $$15-8=7$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 6=-6$$

14. Calculate the sum of the imaginary coefficients.

    $$-20-6=-26$$

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

    $$7-26\times i$$

Answer

The final answer is shown below.