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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 termbyterm 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 termbyterm. 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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 Summer '08
 NOONE
 Calculus, Algebra

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