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11.3 - Add and Subtract Radicals
Worksheet Key 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Worksheet Key 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Worksheet Key 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Worksheet Key 4/12/2019 6:02 PM Add and Subtract Radicals
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Adding and Subtracting Radicals
4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Review Multiply 3x • 2x 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Review Add 3x + 2x 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Steps Reduce radicals to proper simplified form Combine like terms Simplify, if needed 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 1 Add 3√2 + 2√2 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 2 Add 4√8 + 9√8 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Add 3√7 + 9√7 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 2 Add √5 + √20 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 3 Add √5 + √20 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 2 Rationalize 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Rationalize 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Review Solve …NO SOLUTION… right? Not necessarily. 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Steps Imaginary Numbers are numbers that can be written as a real number times “i.” View the radical without the negative and simplify it After simplified form, attach an “i” onto the real number Simplify further, if needed 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 8 Simplify View the radical without the negative and simplify it 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 8 Simplify After simplified form, attach an “i” onto the real number 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 8 Simplify 4/12/2019 6:02 PM Add and Subtract Radicals
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Where does it cross the x-axis? 11.3 - Add and Subtract Radicals
Example 8 Where does it cross the x-axis? +2i x = 0 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 9 Simplify 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 10 Simplify 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Simplify 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Imaginary Numbers Rules with Imaginaries: i1 = i i2 = –1 Answers CAN have an i but CANNOT have two i’s Two i’s make –1 4/12/2019 6:02 PM Add and Subtract Radicals
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How do we cancel the radicals? 11.3 - Add and Subtract Radicals
Why is i2 equal to –1? How do we cancel the radicals? 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 11 Multiply What is another way of writing this problem? 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 12 Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 13 Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 14 Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 15 Solve the equation, x2 = 144 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 16 Solve the equation, 5x = 0 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Solve the equation, 9x = 0 4/12/2019 6:02 PM Add and Subtract Radicals
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SIMPLIFY RADICALS FIRST then OPERATE 11.3 - Add and Subtract Radicals
Complex Numbers Complex Numbers are numbers that are associated with a second real number with an ordered pair a + bi Conjugate is the complex number’s opposite sign Example: i’s conjugate is 2 – 3i Remember: NO IMAGINARY NUMBERS in the denominator SIMPLIFY RADICALS FIRST then OPERATE Real Number Imaginary Number 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 17 Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true. Real parts 4x + 10i = 2 – (4y)i Imaginary parts 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 18 Find the values of x and y that make the equation 2x – 6i = –8 + (20y)i true. 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Find the values of x and y that make the equation true. 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Imaginary Powers Look at the exponent of the imaginary number and divide by 4 View only the remainder or decimal Convert the problem with the number 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Imaginary Powers Imaginary Exponent Remainder Decimal Imaginary Number i1 R1 0.25 i i2 R2 0.50 –1 i3 R3 0.75 –i i4 R0 0.00 1 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Imaginary Powers 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 19 Simplify i7 Imaginary Exponent Remainder Decimal Imaginary Number i1 R1 0.25 i i2 R2 0.50 –1 i3 R3 0.75 –i i4 R0 0.00 1 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 20 Simplify i25 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Simplify i5892 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 21 Simplify i25 + i36 i25 = i i36 = 1 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Simplify i21 + i30 4/12/2019 6:02 PM Add and Subtract Radicals
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Example 22 Rationalize No i ’s AND radicals in the denominator
4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 23 Rationalize 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 24 Rationalize What is the problem with the bottom? We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 25 Rationalize 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Rationalize 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Review Add (2 + 5x) + (3 – 4x) Numbers Variables 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 26 Add (2 + 5i) + (3 – 4i) Real Numbers Imaginary Numbers 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 27 Subtract (7 – 3i) – (5 + 2i) 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Subtract (2 – 4i) – (2 – i) 4/12/2019 6:02 PM Add and Subtract Radicals
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FOIL (First Outer Inner Last) 11.3 - Add and Subtract Radicals
Review Multiply (2 + x)2 FOIL (First Outer Inner Last) 4/12/2019 6:02 PM Add and Subtract Radicals
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FOIL (First Outer Inner Last) 11.3 - Add and Subtract Radicals
Example 28 FOIL (First Outer Inner Last) Multiply (2 + i)2 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Example 29 Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Your Turn Multiply 4/12/2019 6:02 PM Add and Subtract Radicals
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11.3 - Add and Subtract Radicals
Assignment Worksheet 4/12/2019 6:02 PM Add and Subtract Radicals
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