Monitor Projector

# DrDelMath

## Chapter 10: Rational Exponents, Radical, and Complex Numbers

Exercises - Radicals - Exercise Set 10.1
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#### Section 10.2: Rational Exponents

Exercises - Rational Exponents - Exercise Set 10.2
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#### Section 10.3: Simplifying Radical Expressions

The following two facts are quite useful tools when simplifying radicals.

Exercises - Simplifying Radical Expressions - Exercise Set 10.3
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Exercises - Rational Exponents - Exercise Set 10.4
Page 615: Vocabulary Check: 1 - 4. Page 615: Exercises: 1, 2, 3, 5, 6, 9, 11, 12, 27, 29, 35, 36, 39, 40, 45.

#### Section 10.4b:Multiplication of Sums of Radical Expressions

Exercises - Multiplication of Sums of Radical Expressions - Exercise Set 10.4
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#### Section 10.5: Rationalizing Denominators and Numerators of Radical Expressions

Exercises - Rationalizing Denominators and Numerators of Radical Expressions - Exercise Set 10.5
Page 621: Vocabulary Check: 1 - 4. Page 622: Exercises: 1, 3, 5, 13, 17, 19, 23, 25, 35 - 40, 41, 43, 45, 47, 49, 51, 55, 57, 59, 63, 65, 71, 73, 75, 77, 77, 79, 81, 89, 90.

If the radicand of a radical expression contains only numbers, then that radical expression is a number.
If such a radicand appears in an equation it must be treated as any other number would be treated.
For example, to solve the equation $\sqrt{3}x=5$ we simply divide both sides of the equation by $\sqrt{3}$ to obtain the simplest equation $x=\frac{5}{\sqrt{3}}$.
Note the process was no different than if we were to solve 7x = 12.
However, if the radicand of a radical expression contains a variable the situation changes dramatically. In particular, if a radical containing a variable is part of an equation, the solution process must be adjusted to accomodate the radical expression. Such equations are called radical equations. The required process is the subject of this section of the chapter.

It is important to remember that definitions, properties, and processes that are valid for all equations are valid for radical equations. For example, a solution is a number which creates a true statement when substituted into the equation. Of particular importance are the first two properties of equations (listed below) and the definition of equivalent equations (listed below). The definition of equivalent equations and the first two properties of equations allows us to formulate the familiar process for solving linear equations in one variable.

Start with the original equation (the one to be solved) and use the above two properties of equations to generate simpler equations, all equivalent to the original equation, until a simplest equation is obtained whose solution is clearly the number on one side of the equality. Because all equations in the sequence are equivalent to the original, it follows that the solution of the final simplest equation is the solution of the original equation.

This simple process works for all linear equations in one variable but it will not work for radical equations.

It should be clear that in order to solve an equation containing a radical (such as $\sqrt{2x+3}=5$) we must "get rid" of the radical. That is usually accomplished by squaring both sides of the equation, but that seemingly innocent process creates a problem because it does NOT necessarily create an equation which is equivalent to the original (2x + 3 = 25 might not be equivalent to the original equation $\sqrt{2x+3}=5$).

The first of the two facts below alerts us to the potential problem and the second alerts us to a remedy to the problem. The remedy is incorporated into the two Procedure statements.

Exercises - Radical Equations - Exercise Set 10.6
Page 629: Vocabulary Check: 1 - 4. Page 630: Exercises: 1, 3, 5, 7, 15, 17, 19, 21, 27, 31, 39, 43, 53, 55, 57.

#### Section 10.7: Complex Numbers

A more detailed presentation of Complex Numbers is found HERE

Exercises - Complex Numbers - Exercise Set 10.7
Page 639: Vocabulary Check: 1 - 6. Page 640: Exercises: odd numbered problems from 1 to 47. Each problem from 81 to 92.