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TABLE OF CONTENTS
Simplification of Radicals
The Radical Express
Simplifying Radicals with Indices Higher Than 2
The Radical Express with Variable Expressions
Solution to Quadratic Equations by Quadratic Compression
Nature of the Discriminant and Types of Roots
Factoring ax2 + bx + c: The Non-Monic Trinomial Sweep
Solving Factorable Non-Monic Trinomials
Solving Quadratics by the Cool Square Technique
Keys to Quadratic Mastery
Conclusion
The Radical Express with Variable Expressions
Now we go on to a beautiful extension of the Radical Express and show how this method allows you to simplify any radical regardless of the index and regardless of whether you have variable expressions as part of the radicand. Let.s start with a square root radical and then we can extend this idea. Take a look at the following:
√28x3 y2
The method to simplify this is stated as follows:
. treat the number and variables separately;
. simplify the number using the radical express method;
. for the variables, divide the index into the exponent of each variable, placing the whole number quotient as the exponent of the variable outside the radical and the remainder part of the quotient as the exponent of the variable inside the radical. That's it!
Let's demonstrate this technique with √28x3 y2 . As before √28 = 2√7 . Now to do √x3 y2 we divide the index, which is 2 understood (remember the invisible demons) into both the 3 of x3 and the 2 of y2. In the first case we get 1 remainder 1 and in the second we get 1 remainder 0. This means that x will be written with an exponent of 1 outside the radical and an exponent of 1 inside the radical, while y will be written with a 1 outside the radical and 0 inside the radical. Remember that anything to the 0 power is 1, so in effect y will not appear inside the radical.
In other words, √28x3y2 = 2xy√7x. Remember that the xy outside and the x inside all have an understood exponent of 1.
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