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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Exponents and Radicals7
Exponents and Radicals
7.1 Radical Expressions and Functions
7.2 Rational Numbers as Exponents
7.3 Multiplying Radical Expressions
7.4 Dividing Radical Expressions
7.5 Expressions Containing Several Radical Terms
7.6 Solving Radical Equations
7.7 Geometric Applications
The Complex Numbers
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3. Expressions Containing Several Radical Terms
7.5
Expressions Containing Several Radical Terms
Adding and Subtracting Radical Expressions
Products and Quotients of Two or More Radical Terms
Rationalizing Denominators and Numerators (Part 2)
Terms with Differing Indices
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4. Adding and Subtracting Radical Expressions
When two radical expressions have the same indices and radicands, they are said to be like radicals. Like radicals can be combined (added or subtracted) in much the same way that we combined like terms earlier in this text.
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5. Example Simplify by combining like radical terms. Solution
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6. Example Simplify by combining like radical terms. Solution
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7. Products and Quotients of Two or More Radical Terms
Radical expressions often contain factors that have more than one term. Multiplying such expressions is similar to finding products of polynomials. Some products will yield like radical terms, which we can now combine.
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8. Example Multiply. Simplify if possible. Solution
Using the distributive law
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9. Solution
F O I L
F O I L
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10. In part (c) of the last example, notice that the inner and outer products in FOIL are opposites, the result, m – n, is not itself a radical expression. Pairs of radical terms like, are called conjugates.
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11. Rationalizing Denominators and Numerators (Part 2)
The use of conjugates allows us to rationalize denominators or numerators with two terms.
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12. Example Rationalize the denominator: Solution
Multiplying by 1 using the conjugate
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13. To rationalize a numerator with more than one term, we use the conjugate of the numerator.
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14. Terms with Differing Indices
To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation.
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15. To Simplify Products or Quotients with Differing Indices
1. Convert all radical expressions to exponential notation.
2. When the bases are identical, subtract exponents to divide and add exponents to multiply. This may require finding a common denominator.
3. Convert back to radical notation and, if possible, simplify.
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16. Example Multiply and, if possible, simplify: Solution
Converting to exponential notation
Adding exponents
Converting to radical notation
Simplifying
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