DrDelMathIntermediate Algebra

Exercises  Radicals  Exercise Set 10.1
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Exercises  Rational Exponents  Exercise Set 10.2
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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
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Exercises  Multiplication of Sums of Radical Expressions  Exercise Set 10.4
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Exercises  Rationalizing Denominators and Numerators of Radical Expressions  Exercise Set 10.5
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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
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A more detailed presentation of Complex Numbers is found HERE
Exercises  Complex Numbers  Exercise Set 10.7
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Page 640: Exercises: odd numbered problems from 1 to 47. Each problem from 81 to 92.