3332020a07qxd 12605 222 pm page a71 appendix a7 a71

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Unformatted text preview: nvolving Exponents Potential Error x2 3 x2 x3 2x 3 x2 x6 2x 1 x2 Correct Form x5 x2 3 x2 2x 3 x x3 3 2 x 3 x3 3 Comment x6 x2 3 Multiply exponents when raising a power to a power. x5 Add exponents when multiplying powers with like bases. 2 x3 Leave as Exponents have priority over coefficients. 1 x2 x3 . Do not move term-by-term from denominator to numerator. Errors Involving Radicals Potential Error 5x 5 x x2 a2 x x Correct Form 5x 5x a a x Leave as a x2 Comment Radicals apply to every factor inside the radical. a 2. x Leave as Do not apply radicals term-by-term. a. Do not factor minus signs out of square roots. Errors Involving Dividing Out Potential Error a bx a a ax a 1 x 2x 1 bx a x 1 1 x Correct Form a bx a a ax a1 a a a 1 bx a Comment x 1 a x 2x 1 1 2 b x a 1 Divide out common factors, not common terms. x Factor before dividing out. 3 2 Divide out common factors. A good way to avoid errors is to work slowly, write neatly, and talk to yourself. Each time you write a step, ask yourself why the step is algebraically legitimate. You can justify the step below because dividing the numerator and denominator by the same nonzero number produces an equivalent fraction. 2x 6 2x 23 Example 1 x 3 Using the Property for Adding Fractions Describe and correct the error. 1 2x 1 3x 1 5x Solution When adding fractions, use the property for adding fractions: 1 2x 1 3x 3x 2x 6x 2 5x 6x 2 5 6x Now try Exercise 17. 1 a 1 b b a ab . 333202_0A07.qxd 12/6/05 A72 2:22 PM Appendix A Page A72 Review of Fundamental Concepts of Algebra Some Algebra of Calculus In calculus it is often necessary to take a simplified algebraic expression and “unsimplify” it. See the following lists, taken from a standard calculus text. Unusual Factoring Expression 5x 4 8 x2 Useful Calculus Form 54 x 8 3x 6 12 x 6 2x 2 x 3 x x 2 1 12 x 1 12 x Write with fractional coefficient. 3x Write with fractional coefficient. x 2 2 x2 Comment 3 2 Factor out the leading coefficient. 12 1 2 x 2x 1 Factor out factor with lowest power. Writing with Negative Exponents Expression 9 5x3 7 2x Useful Calculus Form 93 x 5 7 2...
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