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Radicals and Rational Exponents Chapter 10

10 Radicals and Rational Exponents 10.1 Finding Roots 10.3 Simplifying Expressions Containing Square Roots 10.4 Simplifying Expressions Containing Higher Roots 10.5 Adding, Subtracting, and Multiplying Radicals 10.6 Dividing Radicals Putting it All Together 10.7 Solving Radical Equations 10.8 Complex Numbers

10.8 Complex Numbers We have seen in previous sections that the square root of a negative number does not exist in the real number system because there is no real number that, when squared, will result in a negative number. For example, is not a real number because there is no real number whose square is -4. The square roots of negative numbers do exist, however, under another system of numbers called complex numbers. Before we define a complex number, we must Define the number i. the number i is called an imaginary number. Note

The following table lists some examples of complex numbers and their real and Imaginary parts. Note Since all real numbers, a, can be written in the form a + 0i, all real numbers are also complex numbers.

Example 1 Solve. Solution Note

Multiply and Divide Square Roots Containing Negative Numbers Note Example 2 Multiply and Simplify. Solution Write each radical in terms of i before multiplying. Multiply Replace with -1.

Add and Subtract Complex Numbers Just as we can add, subtract, and multiply, and divide real numbers, we can perform All of these operations with complex numbers. Example 3 Add or Subtract. Solution Add real parts together and imaginary parts together. Distributive property. Add real parts together and imaginary parts together.

Multiply Complex Numbers We multiply complex numbers just like we would multiply polynomials There may be An additional step, however. Remember to replace with -1. Example 4 Add or Subtract. Solution Distributive property. F O I L You can use FOIL to multiply. Replace with -1. Combine like terms.

Multiply a Complex Number by Its Conjugate The product of a complex number and its conjugate is always a real number. F O I L Replace with -1.

Example 5 Solution Example 6 Solution

Divide Complex Numbers Example 7 Solution Multiply the numerator and denominator by the conjugate of the denominator. Multiply numerators and use to multiply the denominators . Write quotient in the form