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**Complex Numbers** - **Practice Problems**

Perform the indicated operation and write your answer in standard form.**Q.1.**** (4−5i)(12+11i)(4−5i)(12+11i)****Ans. **We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,

All we need to do to finish the problem is to recall that i^{2}=−1. Upon using this fact we can finish the problem.

**Q.2. (−3−i)−(6−7i)****Ans. **We know how to subtract two polynomials and so we also know how to subtract two complex numbers.

**Q.3. (1+4i)−(−16+9i)****Ans. **We know how to subtract two polynomials and so we also know how to subtract two complex numbers.

**Q.4. 8i(10+2i) ****Ans.** We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is distribute the 8i to get,

All we need to do to finish the problem is to recall that i^{2}=−1. Upon using this fact we can finish the problem.

**Q.5. (−3−9i)(1+10i)****Ans. **We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,

All we need to do to finish the problem is to recall that i^{2}=−1. Upon using this fact we can finish the problem.

**Q.6. (2+7i)(8+3i)****Ans. **We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,

All we need to do to finish the problem is to recall that i^{2}=−1. Upon using this fact we can finish the problem.

**Q.7. ****Ans. **Because standard form does not allow for i’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 2−10i.

Multiplying by the conjugate gives,

Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.

**Q.8.** **Ans. **Because standard form does not allow for ii’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 3i.

Multiplying by the conjugate gives,

Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.

**Q.9.****Ans. **Because standard form does not allow for i’s to be in the denominator we’ll need to multiply the numerator and denominator by the conjugate of the denominator, which is 8+i.

Multiplying by the conjugate gives,

Now all we need to do is do the multiplication in the numerator and denominator and put the result in standard form.

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### Revision Notes - Complex Numbers

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- Operations on Complex Numbers
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