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Evaluate \(n^{\text{th}}\)-root radicals.
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Simplify radicals by applying appropriate properties.
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Distinguish between like radicals and unlike radicals.
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Add and subtract like radicals.
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Multiply and divide radicals, simplifying results.
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6.1: Simplify Radical Expressions
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Who doesn’t love the “undo” command on a computer? Radicals are like an undo command to all powers. If taking powers is a forward-action, then radicals require us to think backwards. To prepare for this section, memorize the first twelve perfect squares and the first five cubes. It will be very helpful to recognize powers of 2 and 3.
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6.2: Add and Subtract Radical Expressions
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Rules for radical multiplication and division have a simplicity and ease which lulls students into thinking addition and subtraction will follow suit. However, rules for addition and subtraction have more complication and less flexibility. Adding and subtracting radicals starts with simplifying radicals. We will continue to assume the variables take on nonnegative values only. Let’s explore how gathering like terms can be used with radicals.
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6.3: Rationalize Denominators
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Suppose a fraction a/b contains a radical in the denominator. Rationalizing the denominator is a method of simplification that eliminates radicals from the denominator. The numerator may contain radicals, but we generally don’t worry about that. Only the denominator is rationalized. Sometimes the denominator becomes rational in the simplification steps.