7.5 – Rationalizing the Denominator of Radicals Expressions

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Presentation transcript:

7.5 – Rationalizing the Denominator of Radicals Expressions Radical expressions, at times, are easier to work with if the denominator does not contain a radical. The process to clear the denominator of all radical is referred to as rationalizing the denominator

7.5 – Rationalizing the Denominator of Radicals Expressions

7.5 – Rationalizing the Denominator of Radicals Expressions

7.5 – Rationalizing the Denominator of Radicals Expressions If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required. Review: (x + 3)(x – 3)  x2 – 3x + 3x – 9  x2 – 9 (x + 7)(x – 7)  x2 – 7x + 7x – 49  x2 – 49

7.5 – Rationalizing the Denominator of Radicals Expressions If the denominator contains a radical and it is not a monomial term, then the use of a conjugate is required. conjugate

7.5 – Rationalizing the Denominator of Radicals Expressions conjugate

7.5 – Rationalizing the Denominator of Radicals Expressions conjugate

7.6 – Radical Equations and Problem Solving The Squaring Property of Equality: Examples:

7.6 – Radical Equations and Problem Solving Suggested Guidelines: 1) Isolate the radical to one side of the equation. 2) Square both sides of the equation. 3) Simplify both sides of the equation. 4) Solve for the variable. 5) Check all solutions in the original equation.

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving no solution

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving

7.6 – Radical Equations and Problem Solving

7.7 – Complex Numbers Complex Number System: This system of numbers consists of the set of real numbers and the set of imaginary numbers. Imaginary Unit: The imaginary unit is called i, where and Square roots of a negative number can be written in terms of i.        

Operations with Imaginary Numbers 7.7 – Complex Numbers The imaginary unit is called i, where and Operations with Imaginary Numbers             

7.7 – Complex Numbers The imaginary unit is called i, where and Numbers that can written in the form a + bi, where a and b are real numbers. 3 + 5i 8 – 9i –13 + i The Sum or Difference of Complex Numbers     

7.7 – Complex Numbers      

Multiplying Complex Numbers        

Multiplying Complex Numbers       

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2    

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2    

Dividing Complex Numbers Complex Conjugates: The complex numbers (a + bi) and (a – bi) are complex conjugates of each other and, (a + bi)(a – bi) = a2 + b2    